UniformSampleCone, x

Percentage Accurate: 58.1% → 99.0%

Time: 16.4s

Alternatives: 15

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: 58.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.9%

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

   1. associate-*l* 55.9%

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

   2. +-commutative 55.9%

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

   3. associate-+r- 55.9%

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

   4. fma-def 55.9%

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

   5. +-commutative 55.9%

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

   6. associate-+r- 55.8%

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

   7. fma-def 55.8%

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

3. Simplified 55.8%

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

4. Taylor expanded in ux around -inf 99.2%

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

   1. metadata-eval 99.2%

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

   2. cancel-sign-sub-inv 99.2%

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

   3. *-commutative 99.2%

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

   4. fma-def 99.2%

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

   5. cancel-sign-sub-inv 99.2%

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

   6. metadata-eval 99.2%

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

   7. +-commutative 99.2%

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

   8. *-commutative 99.2%

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

   9. fma-def 99.2%

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

   10. mul-1-neg 99.2%

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

   11. distribute-rgt-neg-in 99.2%

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

   12. unpow2 99.2%

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

   13. mul-1-neg 99.2%

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

   14. sub-neg 99.2%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.9%

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

   1. associate-*l* 55.9%

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

   2. sub-neg 55.9%

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

   3. +-commutative 55.9%

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

   4. distribute-rgt-neg-in 55.9%

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

   5. fma-def 55.9%

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

   6. +-commutative 55.9%

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

   7. associate-+r- 55.9%

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

   8. fma-def 55.9%

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

   9. neg-sub0 55.9%

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

   10. +-commutative 55.9%

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

   11. associate-+r- 55.9%

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

   12. associate--r- 55.9%

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

   13. neg-sub0 55.9%

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

   14. +-commutative 55.9%

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

   15. sub-neg 55.9%

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

   16. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. fma-def 99.2%

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

   2. sub-neg 99.2%

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

   3. metadata-eval 99.2%

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

   4. *-commutative 99.2%

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

   5. unpow2 99.2%

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

   6. +-commutative 99.2%

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

   7. mul-1-neg 99.2%

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

   8. sub-neg 99.2%

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

   9. metadata-eval 99.2%

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

   10. distribute-neg-in 99.2%

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

   11. metadata-eval 99.2%

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

   12. +-commutative 99.2%

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

   13. mul-1-neg 99.2%

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

   14. associate--l+ 99.2%

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

   15. mul-1-neg 99.2%

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

   16. sub-neg 99.2%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.9%

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

   1. associate-*l* 55.9%

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

   2. sub-neg 55.9%

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

   3. +-commutative 55.9%

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

   4. distribute-rgt-neg-in 55.9%

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

   5. fma-def 55.9%

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

   6. +-commutative 55.9%

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

   7. associate-+r- 55.9%

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

   8. fma-def 55.9%

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

   9. neg-sub0 55.9%

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

   10. +-commutative 55.9%

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

   11. associate-+r- 55.9%

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

   12. associate--r- 55.9%

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

   13. neg-sub0 55.9%

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

   14. +-commutative 55.9%

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

   15. sub-neg 55.9%

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

   16. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. fma-def 99.2%

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

   2. sub-neg 99.2%

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

   3. metadata-eval 99.2%

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

   4. *-commutative 99.2%

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

   5. unpow2 99.2%

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

   6. +-commutative 99.2%

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

   7. mul-1-neg 99.2%

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

   8. sub-neg 99.2%

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

   9. metadata-eval 99.2%

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

   10. distribute-neg-in 99.2%

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

   11. metadata-eval 99.2%

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

   12. +-commutative 99.2%

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

   13. mul-1-neg 99.2%

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

   14. associate--l+ 99.2%

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

   15. mul-1-neg 99.2%

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

   16. sub-neg 99.2%

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

6. Simplified 99.2%

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

   1. pow1/2 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)\right)\right)}^{0.5}} \]

   2. count-2 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \color{blue}{\left(2 \cdot \left(1 - maxCos\right)\right)}\right)\right)}^{0.5} \]

8. Applied egg-rr 99.2%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)\right)\right)}^{0.5}} \]

9. Taylor expanded in uy around inf 99.2%

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

    1. *-commutative 99.2%

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

    2. associate-*r* 99.2%

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

    3. *-commutative 99.2%

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

    4. +-commutative 99.2%

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

    5. associate-*r* 99.2%

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

    6. *-commutative 99.2%

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

    7. associate-*r* 99.2%

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

    8. +-commutative 99.2%

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

    9. *-commutative 99.2%

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

    10. associate-*r* 99.2%

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

    11. *-commutative 99.2%

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

    12. distribute-rgt-out 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.9%

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

   1. associate-*l* 55.9%

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

   2. sub-neg 55.9%

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

   3. +-commutative 55.9%

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

   4. distribute-rgt-neg-in 55.9%

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

   5. fma-def 55.9%

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

   6. +-commutative 55.9%

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

   7. associate-+r- 55.9%

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

   8. fma-def 55.9%

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

   9. neg-sub0 55.9%

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

   10. +-commutative 55.9%

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

   11. associate-+r- 55.9%

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

   12. associate--r- 55.9%

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

   13. neg-sub0 55.9%

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

   14. +-commutative 55.9%

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

   15. sub-neg 55.9%

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

   16. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. fma-def 99.2%

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

   2. sub-neg 99.2%

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

   3. metadata-eval 99.2%

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

   4. *-commutative 99.2%

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

   5. unpow2 99.2%

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

   6. +-commutative 99.2%

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

   7. mul-1-neg 99.2%

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

   8. sub-neg 99.2%

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

   9. metadata-eval 99.2%

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

   10. distribute-neg-in 99.2%

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

   11. metadata-eval 99.2%

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

   12. +-commutative 99.2%

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

   13. mul-1-neg 99.2%

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

   14. associate--l+ 99.2%

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

   15. mul-1-neg 99.2%

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

   16. sub-neg 99.2%

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

6. Simplified 99.2%

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

   1. pow1/2 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)\right)\right)}^{0.5}} \]

   2. count-2 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \color{blue}{\left(2 \cdot \left(1 - maxCos\right)\right)}\right)\right)}^{0.5} \]

8. Applied egg-rr 99.2%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)\right)\right)}^{0.5}} \]

9. Taylor expanded in uy around inf 99.2%

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

    1. *-commutative 99.2%

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

    2. associate-*r* 99.2%

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

    3. *-commutative 99.2%

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

    4. +-commutative 99.2%

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

    5. associate-*r* 99.2%

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

    6. *-commutative 99.2%

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

    7. associate-*r* 99.2%

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

    8. +-commutative 99.2%

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

    9. *-commutative 99.2%

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

    10. associate-*r* 99.2%

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

    11. *-commutative 99.2%

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

    12. distribute-rgt-out 99.2%

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

11. Simplified 99.2%

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

    1. pow1 99.2%

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

    2. fma-def 99.2%

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

13. Applied egg-rr 99.2%

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

15. Simplified 99.1%

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

16. Final simplification 99.1%

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

Alternative 5: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0010000000474974513:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0010000000474974513)
   (sqrt (* ux (- (fma maxCos -2.0 2.0) (* ux (pow (- 1.0 maxCos) 2.0)))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0010000000474974513f) {
		tmp = sqrtf((ux * (fmaf(maxCos, -2.0f, 2.0f) - (ux * powf((1.0f - maxCos), 2.0f)))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0010000000474974513))
		tmp = sqrt(Float32(ux * Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) - Float32(ux * (Float32(Float32(1.0) - maxCos) ^ Float32(2.0))))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00100000005

   1. Initial program 56.8%

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

      1. associate-*l* 56.8%

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

      2. +-commutative 56.8%

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

      3. associate-+r- 56.8%

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

      4. fma-def 56.8%

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

      5. +-commutative 56.8%

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

      6. associate-+r- 56.7%

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

      7. fma-def 56.7%

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

   3. Simplified 56.7%

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

   4. Taylor expanded in ux around -inf 99.6%

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

      1. metadata-eval 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-def 99.6%

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

      5. cancel-sign-sub-inv 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. fma-def 99.6%

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

      10. mul-1-neg 99.6%

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

      11. distribute-rgt-neg-in 99.6%

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

      12. unpow2 99.6%

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

      13. mul-1-neg 99.6%

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

      14. sub-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in uy around 0 97.0%

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

      1. fma-def 97.1%

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

      2. +-commutative 97.1%

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

      3. metadata-eval 97.1%

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

      4. cancel-sign-sub-inv 97.1%

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

      5. *-commutative 97.1%

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

      6. sub-neg 97.1%

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

      7. neg-mul-1 97.1%

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

      8. fma-udef 97.0%

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

      9. *-commutative 97.0%

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

      10. fma-udef 97.1%

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

      11. cancel-sign-sub-inv 97.1%

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

      12. metadata-eval 97.1%

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

      13. +-commutative 97.1%

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

      14. fma-def 97.1%

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

      15. mul-1-neg 97.1%

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

   9. Simplified 97.1%

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

   10. Taylor expanded in ux around 0 97.0%

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

       1. +-commutative 97.0%

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

       2. +-commutative 97.0%

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

       3. mul-1-neg 97.0%

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

       4. *-commutative 97.0%

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

       5. distribute-rgt-neg-in 97.0%

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

       6. unpow2 97.0%

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

       7. associate-*r* 97.0%

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

       8. distribute-lft-out 97.0%

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

       9. distribute-rgt-neg-out 97.0%

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

       10. unsub-neg 97.0%

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

       11. *-commutative 97.0%

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

       12. fma-def 97.0%

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

   12. Simplified 97.0%

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

   if 0.00100000005 < uy

   1. Initial program 53.8%

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

      1. associate-*l* 53.8%

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

      2. sub-neg 53.8%

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

      3. +-commutative 53.8%

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

      4. distribute-rgt-neg-in 53.8%

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

      5. fma-def 54.0%

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

      6. +-commutative 54.0%

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

      7. associate-+r- 54.0%

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

      8. fma-def 54.0%

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

      9. neg-sub0 54.0%

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

      10. +-commutative 54.0%

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

      11. associate-+r- 54.0%

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

      12. associate--r- 54.0%

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

      13. neg-sub0 54.0%

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

      14. +-commutative 54.0%

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

      15. sub-neg 54.0%

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

      16. fma-def 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in ux around 0 77.8%

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

   5. Taylor expanded in maxCos around 0 74.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0010000000474974513:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 6: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0010000000474974513:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0010000000474974513)
   (sqrt (* ux (- (fma maxCos -2.0 2.0) (* ux (pow (- 1.0 maxCos) 2.0)))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0010000000474974513f) {
		tmp = sqrtf((ux * (fmaf(maxCos, -2.0f, 2.0f) - (ux * powf((1.0f - maxCos), 2.0f)))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0010000000474974513))
		tmp = sqrt(Float32(ux * Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) - Float32(ux * (Float32(Float32(1.0) - maxCos) ^ Float32(2.0))))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00100000005

   1. Initial program 56.8%

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

      1. associate-*l* 56.8%

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

      2. +-commutative 56.8%

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

      3. associate-+r- 56.8%

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

      4. fma-def 56.8%

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

      5. +-commutative 56.8%

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

      6. associate-+r- 56.7%

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

      7. fma-def 56.7%

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

   3. Simplified 56.7%

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

   4. Taylor expanded in ux around -inf 99.6%

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

      1. metadata-eval 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-def 99.6%

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

      5. cancel-sign-sub-inv 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. fma-def 99.6%

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

      10. mul-1-neg 99.6%

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

      11. distribute-rgt-neg-in 99.6%

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

      12. unpow2 99.6%

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

      13. mul-1-neg 99.6%

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

      14. sub-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in uy around 0 97.0%

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

      1. fma-def 97.1%

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

      2. +-commutative 97.1%

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

      3. metadata-eval 97.1%

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

      4. cancel-sign-sub-inv 97.1%

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

      5. *-commutative 97.1%

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

      6. sub-neg 97.1%

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

      7. neg-mul-1 97.1%

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

      8. fma-udef 97.0%

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

      9. *-commutative 97.0%

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

      10. fma-udef 97.1%

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

      11. cancel-sign-sub-inv 97.1%

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

      12. metadata-eval 97.1%

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

      13. +-commutative 97.1%

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

      14. fma-def 97.1%

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

      15. mul-1-neg 97.1%

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

   9. Simplified 97.1%

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

   10. Taylor expanded in ux around 0 97.0%

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

       1. +-commutative 97.0%

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

       2. +-commutative 97.0%

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

       3. mul-1-neg 97.0%

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

       4. *-commutative 97.0%

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

       5. distribute-rgt-neg-in 97.0%

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

       6. unpow2 97.0%

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

       7. associate-*r* 97.0%

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

       8. distribute-lft-out 97.0%

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

       9. distribute-rgt-neg-out 97.0%

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

       10. unsub-neg 97.0%

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

       11. *-commutative 97.0%

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

       12. fma-def 97.0%

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

   12. Simplified 97.0%

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

   if 0.00100000005 < uy

   1. Initial program 53.8%

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

      1. associate-*l* 53.8%

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

      2. +-commutative 53.8%

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

      3. associate-+r- 53.8%

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

      4. fma-def 53.8%

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

      5. +-commutative 53.8%

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

      6. associate-+r- 53.8%

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

      7. fma-def 53.8%

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

   3. Simplified 53.8%

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

   4. Taylor expanded in ux around 0 77.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0010000000474974513:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 7: 92.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.9%

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

   1. associate-*l* 55.9%

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

   2. +-commutative 55.9%

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

   3. associate-+r- 55.9%

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

   4. fma-def 55.9%

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

   5. +-commutative 55.9%

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

   6. associate-+r- 55.8%

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

   7. fma-def 55.8%

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

3. Simplified 55.8%

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

4. Taylor expanded in ux around -inf 99.2%

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

   1. metadata-eval 99.2%

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

   2. cancel-sign-sub-inv 99.2%

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

   3. *-commutative 99.2%

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

   4. fma-def 99.2%

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

   5. cancel-sign-sub-inv 99.2%

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

   6. metadata-eval 99.2%

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

   7. +-commutative 99.2%

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

   8. *-commutative 99.2%

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

   9. fma-def 99.2%

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

   10. mul-1-neg 99.2%

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

   11. distribute-rgt-neg-in 99.2%

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

   12. unpow2 99.2%

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

   13. mul-1-neg 99.2%

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

   14. sub-neg 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in maxCos around 0 95.0%

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

   1. +-commutative 95.0%

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

   2. neg-mul-1 95.0%

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

   3. unsub-neg 95.0%

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

   4. unpow2 95.0%

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

9. Simplified 95.0%

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

10. Final simplification 95.0%

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

Alternative 8: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0010000000474974513:\\ \;\;\;\;\sqrt{\left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot {ux}^{2}\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0010000000474974513)
   (sqrt
    (+
     (* (- 1.0 maxCos) (* (+ maxCos -1.0) (pow ux 2.0)))
     (* ux (- 2.0 (* 2.0 maxCos)))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0010000000474974513f) {
		tmp = sqrtf((((1.0f - maxCos) * ((maxCos + -1.0f) * powf(ux, 2.0f))) + (ux * (2.0f - (2.0f * maxCos)))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0010000000474974513))
		tmp = sqrt(Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(maxCos + Float32(-1.0)) * (ux ^ Float32(2.0)))) + Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0010000000474974513))
		tmp = sqrt((((single(1.0) - maxCos) * ((maxCos + single(-1.0)) * (ux ^ single(2.0)))) + (ux * (single(2.0) - (single(2.0) * maxCos)))));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00100000005

   1. Initial program 56.8%

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

      1. associate-*l* 56.8%

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

      2. sub-neg 56.8%

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

      3. +-commutative 56.8%

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

      4. distribute-rgt-neg-in 56.8%

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

      5. fma-def 56.8%

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

      6. +-commutative 56.8%

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

      7. associate-+r- 56.7%

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

      8. fma-def 56.7%

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

      9. neg-sub0 56.7%

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

      10. +-commutative 56.7%

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

      11. associate-+r- 56.7%

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

      12. associate--r- 56.7%

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

      13. neg-sub0 56.7%

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

      14. +-commutative 56.7%

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

      15. sub-neg 56.7%

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

      16. fma-def 56.7%

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

   3. Simplified 56.7%

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

   4. Taylor expanded in ux around 0 99.5%

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

      1. fma-def 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. unpow2 99.5%

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

      6. +-commutative 99.5%

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

      7. mul-1-neg 99.5%

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

      8. sub-neg 99.5%

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

      9. metadata-eval 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. +-commutative 99.5%

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

      13. mul-1-neg 99.5%

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

      14. associate--l+ 99.6%

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

      15. mul-1-neg 99.6%

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

      16. sub-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in uy around 0 97.0%

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

   if 0.00100000005 < uy

   1. Initial program 53.8%

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

      1. associate-*l* 53.8%

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

      2. sub-neg 53.8%

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

      3. +-commutative 53.8%

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

      4. distribute-rgt-neg-in 53.8%

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

      5. fma-def 54.0%

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

      6. +-commutative 54.0%

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

      7. associate-+r- 54.0%

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

      8. fma-def 54.0%

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

      9. neg-sub0 54.0%

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

      10. +-commutative 54.0%

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

      11. associate-+r- 54.0%

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

      12. associate--r- 54.0%

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

      13. neg-sub0 54.0%

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

      14. +-commutative 54.0%

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

      15. sub-neg 54.0%

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

      16. fma-def 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in ux around 0 77.8%

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

   5. Taylor expanded in maxCos around 0 74.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0010000000474974513:\\ \;\;\;\;\sqrt{\left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot {ux}^{2}\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 9: 79.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.9%

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

   1. associate-*l* 55.9%

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

   2. sub-neg 55.9%

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

   3. +-commutative 55.9%

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

   4. distribute-rgt-neg-in 55.9%

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

   5. fma-def 55.9%

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

   6. +-commutative 55.9%

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

   7. associate-+r- 55.9%

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

   8. fma-def 55.9%

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

   9. neg-sub0 55.9%

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

   10. +-commutative 55.9%

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

   11. associate-+r- 55.9%

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

   12. associate--r- 55.9%

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

   13. neg-sub0 55.9%

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

   14. +-commutative 55.9%

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

   15. sub-neg 55.9%

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

   16. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. fma-def 99.2%

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

   2. sub-neg 99.2%

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

   3. metadata-eval 99.2%

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

   4. *-commutative 99.2%

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

   5. unpow2 99.2%

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

   6. +-commutative 99.2%

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

   7. mul-1-neg 99.2%

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

   8. sub-neg 99.2%

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

   9. metadata-eval 99.2%

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

   10. distribute-neg-in 99.2%

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

   11. metadata-eval 99.2%

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

   12. +-commutative 99.2%

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

   13. mul-1-neg 99.2%

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

   14. associate--l+ 99.2%

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

   15. mul-1-neg 99.2%

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

   16. sub-neg 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in uy around 0 80.1%

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

8. Final simplification 80.1%

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

Alternative 10: 79.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.9%

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

   1. associate-*l* 55.9%

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

   2. sub-neg 55.9%

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

   3. +-commutative 55.9%

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

   4. distribute-rgt-neg-in 55.9%

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

   5. fma-def 55.9%

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

   6. +-commutative 55.9%

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

   7. associate-+r- 55.9%

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

   8. fma-def 55.9%

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

   9. neg-sub0 55.9%

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

   10. +-commutative 55.9%

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

   11. associate-+r- 55.9%

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

   12. associate--r- 55.9%

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

   13. neg-sub0 55.9%

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

   14. +-commutative 55.9%

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

   15. sub-neg 55.9%

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

   16. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in uy around 0 48.3%

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

5. Taylor expanded in ux around inf 80.1%

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

6. Taylor expanded in maxCos around 0 80.1%

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

   1. associate-*r* 80.1%

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

   2. neg-mul-1 80.1%

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

   3. distribute-rgt1-in 80.1%

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

   4. metadata-eval 80.1%

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

   5. distribute-neg-in 80.1%

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

   6. +-commutative 80.1%

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

   7. distribute-neg-in 80.1%

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

   8. metadata-eval 80.1%

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

   9. sub-neg 80.1%

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

   10. unpow2 80.1%

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

8. Simplified 80.1%

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

9. Final simplification 80.1%

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

Alternative 11: 79.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.9%

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

   1. associate-*l* 55.9%

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

   2. sub-neg 55.9%

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

   3. +-commutative 55.9%

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

   4. distribute-rgt-neg-in 55.9%

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

   5. fma-def 55.9%

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

   6. +-commutative 55.9%

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

   7. associate-+r- 55.9%

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

   8. fma-def 55.9%

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

   9. neg-sub0 55.9%

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

   10. +-commutative 55.9%

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

   11. associate-+r- 55.9%

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

   12. associate--r- 55.9%

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

   13. neg-sub0 55.9%

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

   14. +-commutative 55.9%

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

   15. sub-neg 55.9%

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

   16. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. fma-def 99.2%

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

   2. sub-neg 99.2%

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

   3. metadata-eval 99.2%

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

   4. *-commutative 99.2%

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

   5. unpow2 99.2%

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

   6. +-commutative 99.2%

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

   7. mul-1-neg 99.2%

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

   8. sub-neg 99.2%

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

   9. metadata-eval 99.2%

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

   10. distribute-neg-in 99.2%

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

   11. metadata-eval 99.2%

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

   12. +-commutative 99.2%

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

   13. mul-1-neg 99.2%

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

   14. associate--l+ 99.2%

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

   15. mul-1-neg 99.2%

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

   16. sub-neg 99.2%

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

6. Simplified 99.2%

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

   1. pow1/2 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)\right)\right)}^{0.5}} \]

   2. count-2 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \color{blue}{\left(2 \cdot \left(1 - maxCos\right)\right)}\right)\right)}^{0.5} \]

8. Applied egg-rr 99.2%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)\right)\right)}^{0.5}} \]

9. Taylor expanded in uy around 0 80.1%

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

    1. +-commutative 80.1%

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

    2. associate-*r* 80.1%

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

    3. *-commutative 80.1%

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

    4. associate-*r* 80.1%

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

    5. +-commutative 80.1%

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

    6. *-commutative 80.1%

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

    7. associate-*r* 80.1%

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

    8. *-commutative 80.1%

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

    9. distribute-rgt-out 80.1%

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

    10. *-commutative 80.1%

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

    11. unpow2 80.1%

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

    12. associate-*l* 80.0%

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

    13. sub-neg 80.0%

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

    14. metadata-eval 80.0%

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

11. Simplified 80.0%

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

12. Final simplification 80.0%

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

Alternative 12: 79.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.9%

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

   1. associate-*l* 55.9%

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

   2. sub-neg 55.9%

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

   3. +-commutative 55.9%

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

   4. distribute-rgt-neg-in 55.9%

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

   5. fma-def 55.9%

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

   6. +-commutative 55.9%

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

   7. associate-+r- 55.9%

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

   8. fma-def 55.9%

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

   9. neg-sub0 55.9%

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

   10. +-commutative 55.9%

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

   11. associate-+r- 55.9%

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

   12. associate--r- 55.9%

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

   13. neg-sub0 55.9%

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

   14. +-commutative 55.9%

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

   15. sub-neg 55.9%

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

   16. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. fma-def 99.2%

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

   2. sub-neg 99.2%

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

   3. metadata-eval 99.2%

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

   4. *-commutative 99.2%

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

   5. unpow2 99.2%

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

   6. +-commutative 99.2%

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

   7. mul-1-neg 99.2%

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

   8. sub-neg 99.2%

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

   9. metadata-eval 99.2%

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

   10. distribute-neg-in 99.2%

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

   11. metadata-eval 99.2%

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

   12. +-commutative 99.2%

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

   13. mul-1-neg 99.2%

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

   14. associate--l+ 99.2%

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

   15. mul-1-neg 99.2%

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

   16. sub-neg 99.2%

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

6. Simplified 99.2%

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

   1. pow1/2 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)\right)\right)}^{0.5}} \]

   2. count-2 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot {\left(\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \color{blue}{\left(2 \cdot \left(1 - maxCos\right)\right)}\right)\right)}^{0.5} \]

8. Applied egg-rr 99.2%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{{\left(\mathsf{fma}\left(maxCos + -1, \left(ux \cdot ux\right) \cdot \left(1 - maxCos\right), ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)\right)\right)}^{0.5}} \]

9. Taylor expanded in uy around inf 99.2%

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

    1. *-commutative 99.2%

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

    2. associate-*r* 99.2%

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

    3. *-commutative 99.2%

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

    4. +-commutative 99.2%

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

    5. associate-*r* 99.2%

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

    6. *-commutative 99.2%

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

    7. associate-*r* 99.2%

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

    8. +-commutative 99.2%

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

    9. *-commutative 99.2%

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

    10. associate-*r* 99.2%

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

    11. *-commutative 99.2%

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

    12. distribute-rgt-out 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in uy around 0 80.1%

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

    1. +-commutative 80.1%

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

    2. unpow2 80.1%

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

    3. associate-*r* 80.0%

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

    4. distribute-rgt-out 80.0%

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

    5. sub-neg 80.0%

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

    6. metadata-eval 80.0%

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

14. Simplified 80.0%

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

15. Final simplification 80.0%

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

Alternative 13: 64.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.9%

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

   1. associate-*l* 55.9%

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

   2. +-commutative 55.9%

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

   3. associate-+r- 55.9%

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

   4. fma-def 55.9%

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

   5. +-commutative 55.9%

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

   6. associate-+r- 55.8%

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

   7. fma-def 55.8%

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

3. Simplified 55.8%

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

4. Taylor expanded in ux around -inf 99.2%

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

   1. metadata-eval 99.2%

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

   2. cancel-sign-sub-inv 99.2%

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

   3. *-commutative 99.2%

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

   4. fma-def 99.2%

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

   5. cancel-sign-sub-inv 99.2%

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

   6. metadata-eval 99.2%

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

   7. +-commutative 99.2%

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

   8. *-commutative 99.2%

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

   9. fma-def 99.2%

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

   10. mul-1-neg 99.2%

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

   11. distribute-rgt-neg-in 99.2%

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

   12. unpow2 99.2%

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

   13. mul-1-neg 99.2%

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

   14. sub-neg 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in uy around 0 80.1%

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

   1. fma-def 80.1%

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

   2. +-commutative 80.1%

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

   3. metadata-eval 80.1%

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

   4. cancel-sign-sub-inv 80.1%

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

   5. *-commutative 80.1%

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

   6. sub-neg 80.1%

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

   7. neg-mul-1 80.1%

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

   8. fma-udef 80.1%

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

   9. *-commutative 80.1%

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

   10. fma-udef 80.1%

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

   11. cancel-sign-sub-inv 80.1%

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

   12. metadata-eval 80.1%

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

   13. +-commutative 80.1%

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

   14. fma-def 80.1%

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

   15. mul-1-neg 80.1%

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

9. Simplified 80.1%

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

10. Taylor expanded in ux around 0 65.8%

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

11. Final simplification 65.8%

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

Alternative 14: 75.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.9%

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

   1. associate-*l* 55.9%

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

   2. sub-neg 55.9%

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

   3. +-commutative 55.9%

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

   4. distribute-rgt-neg-in 55.9%

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

   5. fma-def 55.9%

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

   6. +-commutative 55.9%

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

   7. associate-+r- 55.9%

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

   8. fma-def 55.9%

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

   9. neg-sub0 55.9%

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

   10. +-commutative 55.9%

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

   11. associate-+r- 55.9%

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

   12. associate--r- 55.9%

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

   13. neg-sub0 55.9%

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

   14. +-commutative 55.9%

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

   15. sub-neg 55.9%

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

   16. fma-def 55.9%

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

3. Simplified 55.9%

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

4. Taylor expanded in uy around 0 48.3%

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

5. Taylor expanded in ux around inf 80.1%

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

6. Taylor expanded in maxCos around 0 77.3%

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

   1. +-commutative 77.3%

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

   2. mul-1-neg 77.3%

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

   3. unsub-neg 77.3%

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

   4. unpow2 77.3%

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

8. Simplified 77.3%

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

9. Final simplification 77.3%

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

Alternative 15: 61.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.9%

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

   1. associate-*l* 55.9%

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

   2. +-commutative 55.9%

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

   3. associate-+r- 55.9%

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

   4. fma-def 55.9%

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

   5. +-commutative 55.9%

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

   6. associate-+r- 55.8%

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

   7. fma-def 55.8%

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

3. Simplified 55.8%

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

4. Taylor expanded in ux around -inf 99.2%

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

   1. metadata-eval 99.2%

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

   2. cancel-sign-sub-inv 99.2%

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

   3. *-commutative 99.2%

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

   4. fma-def 99.2%

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

   5. cancel-sign-sub-inv 99.2%

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

   6. metadata-eval 99.2%

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

   7. +-commutative 99.2%

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

   8. *-commutative 99.2%

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

   9. fma-def 99.2%

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

   10. mul-1-neg 99.2%

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

   11. distribute-rgt-neg-in 99.2%

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

   12. unpow2 99.2%

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

   13. mul-1-neg 99.2%

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

   14. sub-neg 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in uy around 0 80.1%

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

   1. fma-def 80.1%

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

   2. +-commutative 80.1%

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

   3. metadata-eval 80.1%

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

   4. cancel-sign-sub-inv 80.1%

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

   5. *-commutative 80.1%

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

   6. sub-neg 80.1%

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

   7. neg-mul-1 80.1%

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

   8. fma-udef 80.1%

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

   9. *-commutative 80.1%

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

   10. fma-udef 80.1%

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

   11. cancel-sign-sub-inv 80.1%

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

   12. metadata-eval 80.1%

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

   13. +-commutative 80.1%

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

   14. fma-def 80.1%

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

   15. mul-1-neg 80.1%

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

9. Simplified 80.1%

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

10. Taylor expanded in ux around 0 65.8%

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

11. Taylor expanded in maxCos around 0 63.9%

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

    1. *-commutative 63.9%

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

13. Simplified 63.9%

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

14. Final simplification 63.9%

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

Reproduce

herbie shell --seed 2023238 
(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)))))))