UniformSampleCone, x

Percentage Accurate: 57.5% → 98.9%

Time: 15.8s

Alternatives: 10

Speedup: 3.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.9%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. +-commutative 54.9%

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

   4. distribute-rgt-neg-in 54.9%

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

   5. fma-def 55.0%

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

   6. +-commutative 55.0%

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

   7. associate-+r- 55.0%

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

   8. fma-def 55.0%

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

   9. neg-sub0 55.0%

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

   10. +-commutative 55.0%

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

   11. associate-+r- 54.9%

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

   12. associate--r- 54.9%

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

   13. neg-sub0 54.9%

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

   14. +-commutative 54.9%

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

   15. sub-neg 54.9%

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

   16. fma-def 54.9%

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

3. Simplified 54.9%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. fma-def 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. *-commutative 99.0%

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

   5. unpow2 99.0%

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

   6. +-commutative 99.0%

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

   7. mul-1-neg 99.0%

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

   8. sub-neg 99.0%

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

   9. metadata-eval 99.0%

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

   10. distribute-neg-in 99.0%

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

   11. metadata-eval 99.0%

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

   12. +-commutative 99.0%

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

   13. mul-1-neg 99.0%

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

   14. associate--l+ 99.1%

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

   15. mul-1-neg 99.1%

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

   16. sub-neg 99.1%

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

6. Simplified 99.1%

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

   1. expm1-log1p-u 99.1%

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

8. Applied egg-rr 99.1%

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

9. Taylor expanded in uy around inf 99.1%

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

    1. associate-*r* 99.1%

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

    2. *-commutative 99.1%

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

    3. *-commutative 99.1%

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

    4. *-commutative 99.1%

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

    5. *-commutative 99.1%

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

    6. cancel-sign-sub-inv 99.1%

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

    7. metadata-eval 99.1%

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

    8. distribute-rgt-in 99.1%

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

    9. sub-neg 99.1%

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

    10. *-commutative 99.1%

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

    11. associate-*r* 99.1%

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

    12. *-commutative 99.1%

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

    13. distribute-rgt-out 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in uy around inf 99.2%

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

    1. associate-*r* 99.2%

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

    2. *-commutative 99.2%

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

    3. *-commutative 99.2%

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

    4. sub-neg 99.2%

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

    5. metadata-eval 99.2%

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

    6. unpow2 99.2%

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

    7. fma-udef 99.2%

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

    8. *-commutative 99.2%

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

14. Simplified 99.2%

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

15. Final simplification 99.2%

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

Alternative 2: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(ux \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)\right) \cdot \left(1 - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 9.000000136438757e-5)
   (sqrt (* (* ux (+ 2.0 (* ux (+ maxCos -1.0)))) (- 1.0 maxCos)))
   (* (cos (* uy (* PI 2.0))) (sqrt (- (* 2.0 ux) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 9.000000136438757e-5f) {
		tmp = sqrtf(((ux * (2.0f + (ux * (maxCos + -1.0f)))) * (1.0f - maxCos)));
	} else {
		tmp = cosf((uy * (((float) M_PI) * 2.0f))) * sqrtf(((2.0f * ux) - (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(9.000000136438757e-5))
		tmp = sqrt(Float32(Float32(ux * Float32(Float32(2.0) + Float32(ux * Float32(maxCos + Float32(-1.0))))) * Float32(Float32(1.0) - maxCos)));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(pi) * Float32(2.0)))) * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(9.000000136438757e-5))
		tmp = sqrt(((ux * (single(2.0) + (ux * (maxCos + single(-1.0))))) * (single(1.0) - maxCos)));
	else
		tmp = cos((uy * (single(pi) * single(2.0)))) * sqrt(((single(2.0) * ux) - (ux * ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 9.00000014e-5

   1. Initial program 51.8%

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

      1. associate-*l* 51.8%

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

      2. sub-neg 51.8%

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

      3. +-commutative 51.8%

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

      4. distribute-rgt-neg-in 51.8%

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

      5. fma-def 51.9%

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

      6. +-commutative 51.9%

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

      7. associate-+r- 51.9%

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

      8. fma-def 51.9%

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

      9. neg-sub0 51.9%

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

      10. +-commutative 51.9%

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

      11. associate-+r- 51.8%

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

      12. associate--r- 51.8%

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

      13. neg-sub0 51.8%

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

      14. +-commutative 51.8%

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

      15. sub-neg 51.8%

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

      16. fma-def 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in ux around 0 99.4%

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

      1. fma-def 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

      4. *-commutative 99.4%

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

      5. unpow2 99.4%

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

      6. +-commutative 99.4%

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

      7. mul-1-neg 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. +-commutative 99.4%

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

      13. mul-1-neg 99.4%

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

      14. associate--l+ 99.4%

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

      15. mul-1-neg 99.4%

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

      16. sub-neg 99.4%

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

   6. Simplified 99.4%

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

      1. expm1-log1p-u 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in uy around inf 99.4%

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

       1. associate-*r* 99.4%

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

       2. *-commutative 99.4%

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

       3. *-commutative 99.4%

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

       4. *-commutative 99.4%

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

       5. *-commutative 99.4%

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

       6. cancel-sign-sub-inv 99.4%

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

       7. metadata-eval 99.4%

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

       8. distribute-rgt-in 99.4%

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

       9. sub-neg 99.4%

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

       10. *-commutative 99.4%

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

       11. associate-*r* 99.4%

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

       12. *-commutative 99.4%

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

       13. distribute-rgt-out 99.5%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in uy around 0 99.4%

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

       1. *-commutative 99.4%

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

       2. +-commutative 99.4%

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

       3. *-commutative 99.4%

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

       4. *-commutative 99.4%

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

       5. unpow2 99.4%

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

       6. associate-*l* 99.4%

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

       7. distribute-lft-out 99.4%

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

       8. *-commutative 99.4%

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

       9. sub-neg 99.4%

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

       10. metadata-eval 99.4%

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

   14. Simplified 99.4%

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

   if 9.00000014e-5 < uy

   1. Initial program 59.2%

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

      1. associate-*l* 59.2%

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

      2. +-commutative 59.2%

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

      3. associate-+r- 59.3%

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

      4. fma-def 59.3%

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

      5. +-commutative 59.3%

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

      6. associate-+r- 59.2%

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

      7. fma-def 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in ux around -inf 98.7%

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

      1. metadata-eval 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. *-commutative 98.7%

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

      4. fma-def 98.7%

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

      5. cancel-sign-sub-inv 98.7%

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

      6. metadata-eval 98.7%

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

      7. +-commutative 98.7%

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

      8. *-commutative 98.7%

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

      9. fma-def 98.7%

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

      10. mul-1-neg 98.7%

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

      11. distribute-rgt-neg-in 98.7%

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

      12. unpow2 98.7%

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

      13. mul-1-neg 98.7%

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

      14. sub-neg 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in maxCos around 0 93.4%

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

      1. +-commutative 93.4%

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

      2. neg-mul-1 93.4%

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

      3. unsub-neg 93.4%

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

      4. unpow2 93.4%

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

   9. Simplified 93.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(ux \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)\right) \cdot \left(1 - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \]

Alternative 3: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(ux \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)\right) \cdot \left(1 - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 9.000000136438757e-5)
   (sqrt (* (* ux (+ 2.0 (* ux (+ maxCos -1.0)))) (- 1.0 maxCos)))
   (* (cos (* PI (* 2.0 uy))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 9.000000136438757e-5f) {
		tmp = sqrtf(((ux * (2.0f + (ux * (maxCos + -1.0f)))) * (1.0f - maxCos)));
	} else {
		tmp = cosf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(9.000000136438757e-5))
		tmp = sqrt(Float32(Float32(ux * Float32(Float32(2.0) + Float32(ux * Float32(maxCos + Float32(-1.0))))) * Float32(Float32(1.0) - maxCos)));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * 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 (uy <= single(9.000000136438757e-5))
		tmp = sqrt(((ux * (single(2.0) + (ux * (maxCos + single(-1.0))))) * (single(1.0) - maxCos)));
	else
		tmp = cos((single(pi) * (single(2.0) * uy))) * sqrt((ux * (single(2.0) - ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 9.00000014e-5

   1. Initial program 51.8%

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

      1. associate-*l* 51.8%

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

      2. sub-neg 51.8%

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

      3. +-commutative 51.8%

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

      4. distribute-rgt-neg-in 51.8%

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

      5. fma-def 51.9%

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

      6. +-commutative 51.9%

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

      7. associate-+r- 51.9%

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

      8. fma-def 51.9%

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

      9. neg-sub0 51.9%

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

      10. +-commutative 51.9%

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

      11. associate-+r- 51.8%

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

      12. associate--r- 51.8%

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

      13. neg-sub0 51.8%

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

      14. +-commutative 51.8%

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

      15. sub-neg 51.8%

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

      16. fma-def 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in ux around 0 99.4%

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

      1. fma-def 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

      4. *-commutative 99.4%

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

      5. unpow2 99.4%

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

      6. +-commutative 99.4%

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

      7. mul-1-neg 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. +-commutative 99.4%

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

      13. mul-1-neg 99.4%

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

      14. associate--l+ 99.4%

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

      15. mul-1-neg 99.4%

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

      16. sub-neg 99.4%

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

   6. Simplified 99.4%

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

      1. expm1-log1p-u 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in uy around inf 99.4%

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

       1. associate-*r* 99.4%

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

       2. *-commutative 99.4%

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

       3. *-commutative 99.4%

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

       4. *-commutative 99.4%

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

       5. *-commutative 99.4%

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

       6. cancel-sign-sub-inv 99.4%

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

       7. metadata-eval 99.4%

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

       8. distribute-rgt-in 99.4%

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

       9. sub-neg 99.4%

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

       10. *-commutative 99.4%

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

       11. associate-*r* 99.4%

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

       12. *-commutative 99.4%

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

       13. distribute-rgt-out 99.5%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in uy around 0 99.4%

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

       1. *-commutative 99.4%

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

       2. +-commutative 99.4%

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

       3. *-commutative 99.4%

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

       4. *-commutative 99.4%

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

       5. unpow2 99.4%

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

       6. associate-*l* 99.4%

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

       7. distribute-lft-out 99.4%

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

       8. *-commutative 99.4%

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

       9. sub-neg 99.4%

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

       10. metadata-eval 99.4%

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

   14. Simplified 99.4%

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

   if 9.00000014e-5 < uy

   1. Initial program 59.2%

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

      1. associate-*l* 59.2%

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

      2. sub-neg 59.2%

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

      3. +-commutative 59.2%

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

      4. distribute-rgt-neg-in 59.2%

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

      5. fma-def 59.2%

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

      6. +-commutative 59.2%

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

      7. associate-+r- 59.3%

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

      8. fma-def 59.3%

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

      9. neg-sub0 59.3%

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

      10. +-commutative 59.3%

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

      11. associate-+r- 59.2%

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

      12. associate--r- 59.2%

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

      13. neg-sub0 59.2%

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

      14. +-commutative 59.2%

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

      15. sub-neg 59.2%

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

      16. fma-def 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in ux around 0 98.6%

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

      1. fma-def 98.6%

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

      2. sub-neg 98.6%

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

      3. metadata-eval 98.6%

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

      4. *-commutative 98.6%

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

      5. unpow2 98.6%

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

      6. +-commutative 98.6%

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

      7. mul-1-neg 98.6%

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

      8. sub-neg 98.6%

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

      9. metadata-eval 98.6%

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

      10. distribute-neg-in 98.6%

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

      11. metadata-eval 98.6%

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

      12. +-commutative 98.6%

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

      13. mul-1-neg 98.6%

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

      14. associate--l+ 98.7%

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

      15. mul-1-neg 98.7%

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

      16. sub-neg 98.7%

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

   6. Simplified 98.7%

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

      1. expm1-log1p-u 98.5%

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

   8. Applied egg-rr 98.5%

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

   9. Taylor expanded in uy around inf 98.7%

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

       1. associate-*r* 98.7%

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

       2. *-commutative 98.7%

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

       3. *-commutative 98.7%

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

       4. *-commutative 98.7%

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

       5. *-commutative 98.7%

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

       6. cancel-sign-sub-inv 98.7%

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

       7. metadata-eval 98.7%

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

       8. distribute-rgt-in 98.7%

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

       9. sub-neg 98.7%

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

       10. *-commutative 98.7%

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

       11. associate-*r* 98.7%

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

       12. *-commutative 98.7%

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

       13. distribute-rgt-out 98.7%

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

   11. Simplified 98.7%

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

   12. Taylor expanded in maxCos around 0 93.4%

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

       1. associate-*r* 93.4%

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

       2. *-commutative 93.4%

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

       3. +-commutative 93.4%

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

       4. unpow2 93.4%

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

       5. associate-*r* 93.4%

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

       6. distribute-rgt-out 93.4%

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

       7. mul-1-neg 93.4%

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

   14. Simplified 93.4%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(ux \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)\right) \cdot \left(1 - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

Alternative 4: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00419999985024333:\\ \;\;\;\;\sqrt{\left(ux \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)\right) \cdot \left(1 - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00419999985024333)
   (sqrt (* (* ux (+ 2.0 (* ux (+ maxCos -1.0)))) (- 1.0 maxCos)))
   (* (cos (* uy (* PI 2.0))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00419999985024333f) {
		tmp = sqrtf(((ux * (2.0f + (ux * (maxCos + -1.0f)))) * (1.0f - maxCos)));
	} else {
		tmp = cosf((uy * (((float) M_PI) * 2.0f))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00419999985024333))
		tmp = sqrt(Float32(Float32(ux * Float32(Float32(2.0) + Float32(ux * Float32(maxCos + Float32(-1.0))))) * Float32(Float32(1.0) - maxCos)));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(pi) * Float32(2.0)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.00419999985024333))
		tmp = sqrt(((ux * (single(2.0) + (ux * (maxCos + single(-1.0))))) * (single(1.0) - maxCos)));
	else
		tmp = cos((uy * (single(pi) * single(2.0)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00419999985

   1. Initial program 54.2%

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

      1. associate-*l* 54.2%

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

      2. sub-neg 54.2%

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

      3. +-commutative 54.2%

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

      4. distribute-rgt-neg-in 54.2%

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

      5. fma-def 54.4%

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

      6. +-commutative 54.4%

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

      7. associate-+r- 54.4%

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

      8. fma-def 54.4%

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

      9. neg-sub0 54.4%

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

      10. +-commutative 54.4%

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

      11. associate-+r- 54.3%

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

      12. associate--r- 54.3%

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

      13. neg-sub0 54.3%

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

      14. +-commutative 54.3%

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

      15. sub-neg 54.3%

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

      16. fma-def 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in ux around 0 99.3%

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

      1. fma-def 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. *-commutative 99.3%

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

      5. unpow2 99.3%

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

      6. +-commutative 99.3%

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

      7. mul-1-neg 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. +-commutative 99.3%

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

      13. mul-1-neg 99.3%

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

      14. associate--l+ 99.3%

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

      15. mul-1-neg 99.3%

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

      16. sub-neg 99.3%

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

   6. Simplified 99.3%

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

      1. expm1-log1p-u 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in uy around inf 99.3%

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

       1. associate-*r* 99.3%

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

       2. *-commutative 99.3%

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

       3. *-commutative 99.3%

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

       4. *-commutative 99.3%

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

       5. *-commutative 99.3%

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

       6. cancel-sign-sub-inv 99.3%

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

       7. metadata-eval 99.3%

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

       8. distribute-rgt-in 99.3%

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

       9. sub-neg 99.3%

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

       10. *-commutative 99.3%

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

       11. associate-*r* 99.3%

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

       12. *-commutative 99.3%

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

       13. distribute-rgt-out 99.3%

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

   11. Simplified 99.3%

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

   12. Taylor expanded in uy around 0 95.3%

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

       1. *-commutative 95.3%

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

       2. +-commutative 95.3%

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

       3. *-commutative 95.3%

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

       4. *-commutative 95.3%

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

       5. unpow2 95.3%

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

       6. associate-*l* 95.3%

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

       7. distribute-lft-out 95.3%

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

       8. *-commutative 95.3%

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

       9. sub-neg 95.3%

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

       10. metadata-eval 95.3%

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

   14. Simplified 95.3%

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

   if 0.00419999985 < uy

   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. +-commutative 57.1%

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

      3. associate-+r- 57.4%

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

      4. fma-def 57.4%

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

      5. +-commutative 57.4%

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

      6. associate-+r- 57.2%

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

      7. fma-def 57.2%

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

   3. Simplified 57.2%

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

   4. Taylor expanded in ux around 0 45.5%

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

   5. Taylor expanded in maxCos around 0 71.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.00419999985024333:\\ \;\;\;\;\sqrt{\left(ux \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)\right) \cdot \left(1 - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 5: 79.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.9%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. +-commutative 54.9%

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

   4. distribute-rgt-neg-in 54.9%

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

   5. fma-def 55.0%

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

   6. +-commutative 55.0%

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

   7. associate-+r- 55.0%

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

   8. fma-def 55.0%

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

   9. neg-sub0 55.0%

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

   10. +-commutative 55.0%

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

   11. associate-+r- 54.9%

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

   12. associate--r- 54.9%

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

   13. neg-sub0 54.9%

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

   14. +-commutative 54.9%

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

   15. sub-neg 54.9%

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

   16. fma-def 54.9%

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

3. Simplified 54.9%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. fma-def 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. *-commutative 99.0%

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

   5. unpow2 99.0%

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

   6. +-commutative 99.0%

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

   7. mul-1-neg 99.0%

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

   8. sub-neg 99.0%

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

   9. metadata-eval 99.0%

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

   10. distribute-neg-in 99.0%

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

   11. metadata-eval 99.0%

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

   12. +-commutative 99.0%

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

   13. mul-1-neg 99.0%

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

   14. associate--l+ 99.1%

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

   15. mul-1-neg 99.1%

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

   16. sub-neg 99.1%

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

6. Simplified 99.1%

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

   1. expm1-log1p-u 99.1%

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

8. Applied egg-rr 99.1%

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

9. Taylor expanded in uy around inf 99.1%

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

    1. associate-*r* 99.1%

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

    2. *-commutative 99.1%

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

    3. *-commutative 99.1%

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

    4. *-commutative 99.1%

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

    5. *-commutative 99.1%

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

    6. cancel-sign-sub-inv 99.1%

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

    7. metadata-eval 99.1%

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

    8. distribute-rgt-in 99.1%

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

    9. sub-neg 99.1%

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

    10. *-commutative 99.1%

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

    11. associate-*r* 99.1%

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

    12. *-commutative 99.1%

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

    13. distribute-rgt-out 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in uy around 0 80.6%

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

    1. *-commutative 80.6%

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

    2. +-commutative 80.6%

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

    3. *-commutative 80.6%

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

    4. *-commutative 80.6%

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

    5. unpow2 80.6%

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

    6. associate-*l* 80.6%

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

    7. distribute-lft-out 80.6%

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

    8. *-commutative 80.6%

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

    9. sub-neg 80.6%

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

    10. metadata-eval 80.6%

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

14. Simplified 80.6%

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

15. Final simplification 80.6%

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

Alternative 6: 73.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0003000000142492354:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(ux + -1\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0003000000142492354)
   (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
   (sqrt (+ 1.0 (* (+ ux -1.0) (- 1.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0003000000142492354f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((ux + -1.0f) * (1.0f - ux))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.0003000000142492354e0) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((ux + (-1.0e0)) * (1.0e0 - ux))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0003000000142492354))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(ux + Float32(-1.0)) * Float32(Float32(1.0) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.0003000000142492354))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((ux + single(-1.0)) * (single(1.0) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 3.00000014e-4

   1. Initial program 36.6%

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

      1. add-cube-cbrt 36.6%

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

      2. pow3 36.6%

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

      3. associate-*r* 36.6%

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

      4. pow2 36.6%

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

      5. +-commutative 36.6%

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

      6. fma-def 36.6%

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

   3. Applied egg-rr 36.6%

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

   4. Taylor expanded in uy around 0 33.3%

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

      1. associate-+r- 33.3%

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

   6. Simplified 33.3%

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

   7. Taylor expanded in ux around 0 76.4%

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

   if 3.00000014e-4 < ux

   1. Initial program 88.3%

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

      1. associate-*l* 88.3%

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

      2. sub-neg 88.3%

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

      3. +-commutative 88.3%

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

      4. distribute-rgt-neg-in 88.3%

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

      5. fma-def 88.6%

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

      6. +-commutative 88.6%

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

      7. associate-+r- 88.4%

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

      8. fma-def 88.4%

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

      9. neg-sub0 88.4%

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

      10. +-commutative 88.4%

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

      11. associate-+r- 88.3%

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

      12. associate--r- 88.3%

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

      13. neg-sub0 88.3%

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

      14. +-commutative 88.3%

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

      15. sub-neg 88.3%

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

      16. fma-def 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in uy around 0 73.0%

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

   5. Taylor expanded in maxCos around 0 70.1%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0003000000142492354:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(ux + -1\right) \cdot \left(1 - ux\right)}\\ \end{array} \]

Alternative 7: 78.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.9%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. +-commutative 54.9%

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

   4. distribute-rgt-neg-in 54.9%

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

   5. fma-def 55.0%

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

   6. +-commutative 55.0%

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

   7. associate-+r- 55.0%

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

   8. fma-def 55.0%

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

   9. neg-sub0 55.0%

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

   10. +-commutative 55.0%

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

   11. associate-+r- 54.9%

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

   12. associate--r- 54.9%

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

   13. neg-sub0 54.9%

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

   14. +-commutative 54.9%

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

   15. sub-neg 54.9%

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

   16. fma-def 54.9%

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

3. Simplified 54.9%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. fma-def 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. *-commutative 99.0%

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

   5. unpow2 99.0%

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

   6. +-commutative 99.0%

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

   7. mul-1-neg 99.0%

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

   8. sub-neg 99.0%

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

   9. metadata-eval 99.0%

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

   10. distribute-neg-in 99.0%

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

   11. metadata-eval 99.0%

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

   12. +-commutative 99.0%

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

   13. mul-1-neg 99.0%

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

   14. associate--l+ 99.1%

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

   15. mul-1-neg 99.1%

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

   16. sub-neg 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in uy around 0 80.6%

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

8. Taylor expanded in maxCos around 0 79.7%

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

   1. mul-1-neg 79.7%

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

   2. unpow2 79.7%

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

   3. distribute-rgt-neg-in 79.7%

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

10. Simplified 79.7%

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

11. Final simplification 79.7%

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

Alternative 8: 64.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.9%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. +-commutative 54.9%

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

   4. distribute-rgt-neg-in 54.9%

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

   5. fma-def 55.0%

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

   6. +-commutative 55.0%

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

   7. associate-+r- 55.0%

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

   8. fma-def 55.0%

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

   9. neg-sub0 55.0%

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

   10. +-commutative 55.0%

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

   11. associate-+r- 54.9%

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

   12. associate--r- 54.9%

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

   13. neg-sub0 54.9%

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

   14. +-commutative 54.9%

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

   15. sub-neg 54.9%

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

   16. fma-def 54.9%

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

3. Simplified 54.9%

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

4. Taylor expanded in ux around 0 78.2%

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

5. Taylor expanded in uy around 0 67.0%

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

   1. neg-mul-1 67.0%

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

   2. unsub-neg 67.0%

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

   3. sub-neg 67.0%

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

   4. metadata-eval 67.0%

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

   5. +-commutative 67.0%

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

   6. associate--r+ 67.0%

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

   7. metadata-eval 67.0%

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

7. Simplified 67.0%

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

8. Final simplification 67.0%

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

Alternative 9: 64.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.9%

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

   1. add-cube-cbrt 54.8%

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

   2. pow3 54.8%

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

   3. associate-*r* 54.8%

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

   4. pow2 54.8%

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

   5. +-commutative 54.8%

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

   6. fma-def 54.8%

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

3. Applied egg-rr 54.8%

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

4. Taylor expanded in uy around 0 47.4%

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

   1. associate-+r- 47.6%

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

6. Simplified 47.6%

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

7. Taylor expanded in ux around 0 67.0%

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

8. Final simplification 67.0%

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

Alternative 10: 61.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.9%

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

   1. add-cube-cbrt 54.8%

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

   2. pow3 54.8%

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

   3. associate-*r* 54.8%

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

   4. pow2 54.8%

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

   5. +-commutative 54.8%

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

   6. fma-def 54.8%

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

3. Applied egg-rr 54.8%

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

4. Taylor expanded in uy around 0 47.4%

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

   1. associate-+r- 47.6%

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

6. Simplified 47.6%

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

7. Taylor expanded in ux around 0 67.0%

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

8. Taylor expanded in maxCos around 0 64.6%

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

9. Final simplification 64.6%

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

Reproduce

herbie shell --seed 2023257 
(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)))))))