UniformSampleCone, x

Percentage Accurate: 57.7% → 98.9%

Time: 14.0s

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: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.9%

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

   4. fma-def 52.9%

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

   5. +-commutative 52.9%

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

   6. associate-+r- 52.8%

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

   7. fma-def 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in ux around -inf 56.8%

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

   1. +-commutative 56.8%

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

   2. mul-1-neg 56.8%

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

   3. unsub-neg 56.8%

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

   4. unpow2 56.8%

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

   5. mul-1-neg 56.8%

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

   6. unsub-neg 56.8%

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

   7. +-commutative 56.8%

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

   8. *-commutative 56.8%

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

   9. fma-def 56.8%

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

6. Simplified 56.8%

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

7. Taylor expanded in maxCos around 0 99.0%

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

   1. distribute-lft-out-- 99.0%

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

   2. associate-*r* 99.0%

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

   3. metadata-eval 99.0%

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

   4. distribute-rgt-neg-in 99.0%

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

   5. distribute-lft-neg-in 99.0%

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

   6. neg-mul-1 99.0%

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

   7. associate-*r* 99.0%

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

   8. distribute-lft-out-- 99.0%

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

   9. associate-*r* 99.0%

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

   10. associate-+l+ 99.0%

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

   11. +-commutative 99.0%

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

9. Simplified 99.0%

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

    1. associate-*r* 99.0%

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

    2. add-cbrt-cube 99.0%

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

    3. add-cbrt-cube 99.0%

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

    4. cbrt-unprod 99.0%

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

    5. pow3 99.0%

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

    6. pow3 99.0%

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

11. Applied egg-rr 99.0%

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

12. Final simplification 99.0%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.9%

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

   4. fma-def 52.9%

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

   5. +-commutative 52.9%

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

   6. associate-+r- 52.8%

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

   7. fma-def 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in ux around -inf 56.8%

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

   1. +-commutative 56.8%

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

   2. mul-1-neg 56.8%

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

   3. unsub-neg 56.8%

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

   4. unpow2 56.8%

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

   5. mul-1-neg 56.8%

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

   6. unsub-neg 56.8%

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

   7. +-commutative 56.8%

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

   8. *-commutative 56.8%

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

   9. fma-def 56.8%

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

6. Simplified 56.8%

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

7. Taylor expanded in maxCos around 0 99.0%

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

   1. distribute-lft-out-- 99.0%

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

   2. associate-*r* 99.0%

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

   3. metadata-eval 99.0%

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

   4. distribute-rgt-neg-in 99.0%

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

   5. distribute-lft-neg-in 99.0%

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

   6. neg-mul-1 99.0%

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

   7. associate-*r* 99.0%

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

   8. distribute-lft-out-- 99.0%

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

   9. associate-*r* 99.0%

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

   10. associate-+l+ 99.0%

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

   11. +-commutative 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Alternative 3: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.9%

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

   4. fma-def 52.9%

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

   5. +-commutative 52.9%

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

   6. associate-+r- 52.8%

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

   7. fma-def 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in ux around -inf 56.8%

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

   1. +-commutative 56.8%

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

   2. mul-1-neg 56.8%

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

   3. unsub-neg 56.8%

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

   4. unpow2 56.8%

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

   5. mul-1-neg 56.8%

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

   6. unsub-neg 56.8%

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

   7. +-commutative 56.8%

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

   8. *-commutative 56.8%

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

   9. fma-def 56.8%

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

6. Simplified 56.8%

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

7. Taylor expanded in maxCos around 0 98.0%

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

   1. +-commutative 98.0%

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

   2. associate--l+ 98.0%

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

   3. distribute-lft-out-- 98.0%

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

   4. associate-*r* 98.0%

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

   5. unpow2 98.0%

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

   6. unpow2 98.0%

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

   7. distribute-rgt-out-- 98.0%

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

9. Simplified 98.0%

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

10. Final simplification 98.0%

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

Alternative 4: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) - \left(ux \cdot ux\right) \cdot {\left(1 - maxCos\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00019999999494757503)
   (sqrt
    (- (* ux (fma maxCos -2.0 2.0)) (* (* ux ux) (pow (- 1.0 maxCos) 2.0))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (- (* 2.0 ux) (* ux ux))))))

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if uy < 1.99999995e-4

   1. Initial program 52.5%

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

      1. associate-*l* 52.5%

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

      2. +-commutative 52.5%

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

      3. associate-+r- 52.6%

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

      4. fma-def 52.6%

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

      5. +-commutative 52.6%

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

      6. associate-+r- 52.5%

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

      7. fma-def 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in ux around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

      4. fma-def 99.4%

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

      5. cancel-sign-sub-inv 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

      8. *-commutative 99.4%

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

      9. fma-def 99.4%

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

      10. mul-1-neg 99.4%

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

      11. *-commutative 99.4%

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

      12. distribute-rgt-neg-in 99.4%

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

      13. mul-1-neg 99.4%

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

      14. unsub-neg 99.4%

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

      15. unpow2 99.4%

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

      16. distribute-rgt-neg-in 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in uy around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. associate-*r* 98.9%

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

      3. sub-neg 98.9%

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

      4. mul-1-neg 98.9%

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

      5. associate-*r* 98.9%

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

      6. mul-1-neg 98.9%

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

      7. unsub-neg 98.9%

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

      8. +-commutative 98.9%

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

      9. *-commutative 98.9%

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

      10. fma-udef 98.9%

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

      11. unpow2 98.9%

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

      12. mul-1-neg 98.9%

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

      13. sub-neg 98.9%

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

   9. Simplified 98.9%

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

   if 1.99999995e-4 < uy

   1. Initial program 53.3%

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

      1. associate-*l* 53.3%

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

      2. +-commutative 53.3%

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

      3. associate-+r- 53.4%

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

      4. fma-def 53.4%

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

      5. +-commutative 53.4%

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

      6. associate-+r- 53.3%

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

      7. fma-def 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in ux around -inf 58.0%

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

      1. +-commutative 58.0%

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

      2. mul-1-neg 58.0%

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

      3. unsub-neg 58.0%

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

      4. unpow2 58.0%

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

      5. mul-1-neg 58.0%

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

      6. unsub-neg 58.0%

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

      7. +-commutative 58.0%

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

      8. *-commutative 58.0%

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

      9. fma-def 58.0%

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

   6. Simplified 58.0%

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

   7. Taylor expanded in maxCos around 0 56.6%

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

   8. Taylor expanded in maxCos around 0 93.2%

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

      1. *-commutative 93.2%

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

      2. unpow2 93.2%

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

   10. Simplified 93.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) - \left(ux \cdot ux\right) \cdot {\left(1 - maxCos\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \]

Alternative 5: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos \cdot \left(ux - ux \cdot ux\right), ux \cdot \left(2 - ux\right)\right) - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00019999999494757503)
   (sqrt
    (-
     (fma -2.0 (* maxCos (- ux (* ux ux))) (* ux (- 2.0 ux)))
     (* (* ux ux) (* maxCos maxCos))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (- (* 2.0 ux) (* ux ux))))))

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if uy < 1.99999995e-4

   1. Initial program 52.5%

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

      1. associate-*l* 52.5%

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

      2. +-commutative 52.5%

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

      3. associate-+r- 52.6%

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

      4. fma-def 52.6%

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

      5. +-commutative 52.6%

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

      6. associate-+r- 52.5%

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

      7. fma-def 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in ux around -inf 56.2%

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

      1. +-commutative 56.2%

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

      2. mul-1-neg 56.2%

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

      3. unsub-neg 56.2%

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

      4. unpow2 56.2%

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

      5. mul-1-neg 56.2%

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

      6. unsub-neg 56.2%

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

      7. +-commutative 56.2%

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

      8. *-commutative 56.2%

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

      9. fma-def 56.2%

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

   6. Simplified 56.2%

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

   7. Taylor expanded in maxCos around 0 99.4%

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

      1. distribute-lft-out-- 99.4%

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

      2. associate-*r* 99.4%

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

      3. metadata-eval 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

      5. distribute-lft-neg-in 99.4%

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

      6. neg-mul-1 99.4%

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

      7. associate-*r* 99.4%

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

      8. distribute-lft-out-- 99.4%

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

      9. associate-*r* 99.4%

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

      10. associate-+l+ 99.4%

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

      11. +-commutative 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in uy around 0 98.8%

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

       1. fma-def 98.8%

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

       2. unpow2 98.8%

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

       3. *-commutative 98.8%

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

       4. unpow2 98.8%

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

       5. unpow2 98.8%

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

   12. Simplified 98.8%

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

   if 1.99999995e-4 < uy

   1. Initial program 53.3%

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

      1. associate-*l* 53.3%

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

      2. +-commutative 53.3%

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

      3. associate-+r- 53.4%

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

      4. fma-def 53.4%

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

      5. +-commutative 53.4%

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

      6. associate-+r- 53.3%

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

      7. fma-def 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in ux around -inf 58.0%

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

      1. +-commutative 58.0%

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

      2. mul-1-neg 58.0%

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

      3. unsub-neg 58.0%

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

      4. unpow2 58.0%

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

      5. mul-1-neg 58.0%

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

      6. unsub-neg 58.0%

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

      7. +-commutative 58.0%

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

      8. *-commutative 58.0%

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

      9. fma-def 58.0%

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

   6. Simplified 58.0%

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

   7. Taylor expanded in maxCos around 0 56.6%

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

   8. Taylor expanded in maxCos around 0 93.2%

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

      1. *-commutative 93.2%

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

      2. unpow2 93.2%

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

   10. Simplified 93.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos \cdot \left(ux - ux \cdot ux\right), ux \cdot \left(2 - ux\right)\right) - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \]

Alternative 6: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(2 - ux\right)\\ \mathbf{if}\;uy \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos \cdot \left(ux - ux \cdot ux\right), t_0\right) - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 2.0 ux))))
   (if (<= uy 0.00019999999494757503)
     (sqrt
      (-
       (fma -2.0 (* maxCos (- ux (* ux ux))) t_0)
       (* (* ux ux) (* maxCos maxCos))))
     (* (cos (* uy (* 2.0 PI))) (sqrt t_0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (2.0f - ux);
	float tmp;
	if (uy <= 0.00019999999494757503f) {
		tmp = sqrtf((fmaf(-2.0f, (maxCos * (ux - (ux * ux))), t_0) - ((ux * ux) * (maxCos * maxCos))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf(t_0);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(2.0) - ux))
	tmp = Float32(0.0)
	if (uy <= Float32(0.00019999999494757503))
		tmp = sqrt(Float32(fma(Float32(-2.0), Float32(maxCos * Float32(ux - Float32(ux * ux))), t_0) - Float32(Float32(ux * ux) * Float32(maxCos * maxCos))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 1.99999995e-4

   1. Initial program 52.5%

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

      1. associate-*l* 52.5%

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

      2. +-commutative 52.5%

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

      3. associate-+r- 52.6%

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

      4. fma-def 52.6%

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

      5. +-commutative 52.6%

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

      6. associate-+r- 52.5%

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

      7. fma-def 52.5%

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

   3. Simplified 52.5%

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

   4. Taylor expanded in ux around -inf 56.2%

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

      1. +-commutative 56.2%

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

      2. mul-1-neg 56.2%

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

      3. unsub-neg 56.2%

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

      4. unpow2 56.2%

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

      5. mul-1-neg 56.2%

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

      6. unsub-neg 56.2%

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

      7. +-commutative 56.2%

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

      8. *-commutative 56.2%

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

      9. fma-def 56.2%

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

   6. Simplified 56.2%

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

   7. Taylor expanded in maxCos around 0 99.4%

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

      1. distribute-lft-out-- 99.4%

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

      2. associate-*r* 99.4%

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

      3. metadata-eval 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

      5. distribute-lft-neg-in 99.4%

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

      6. neg-mul-1 99.4%

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

      7. associate-*r* 99.4%

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

      8. distribute-lft-out-- 99.4%

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

      9. associate-*r* 99.4%

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

      10. associate-+l+ 99.4%

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

      11. +-commutative 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in uy around 0 98.8%

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

       1. fma-def 98.8%

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

       2. unpow2 98.8%

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

       3. *-commutative 98.8%

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

       4. unpow2 98.8%

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

       5. unpow2 98.8%

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

   12. Simplified 98.8%

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

   if 1.99999995e-4 < uy

   1. Initial program 53.3%

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

      1. associate-*l* 53.3%

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

      2. +-commutative 53.3%

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

      3. associate-+r- 53.4%

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

      4. fma-def 53.4%

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

      5. +-commutative 53.4%

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

      6. associate-+r- 53.3%

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

      7. fma-def 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in ux around -inf 98.2%

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

      1. +-commutative 98.2%

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

      2. metadata-eval 98.2%

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

      3. cancel-sign-sub-inv 98.2%

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

      4. fma-def 98.2%

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

      5. cancel-sign-sub-inv 98.2%

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

      6. metadata-eval 98.2%

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

      7. +-commutative 98.2%

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

      8. *-commutative 98.2%

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

      9. fma-def 98.2%

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

      10. mul-1-neg 98.2%

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

      11. *-commutative 98.2%

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

      12. distribute-rgt-neg-in 98.2%

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

      13. mul-1-neg 98.2%

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

      14. unsub-neg 98.2%

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

      15. unpow2 98.2%

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

      16. distribute-rgt-neg-in 98.2%

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

   6. Simplified 98.2%

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

      1. associate-*r* 98.3%

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

      2. add-cbrt-cube 98.3%

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

      3. add-cbrt-cube 98.3%

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

      4. cbrt-unprod 98.3%

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

      5. pow3 98.3%

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

      6. pow3 98.3%

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

   8. Applied egg-rr 98.3%

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

   9. Taylor expanded in maxCos around 0 93.2%

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

       1. *-commutative 93.2%

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

       2. associate-*r* 93.2%

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

       3. mul-1-neg 93.2%

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

       4. +-commutative 93.2%

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

       5. sub-neg 93.2%

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

       6. unpow2 93.2%

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

       7. distribute-rgt-out-- 93.2%

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

   11. Simplified 93.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos \cdot \left(ux - ux \cdot ux\right), ux \cdot \left(2 - ux\right)\right) - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

Alternative 7: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0005000000237487257:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos \cdot \left(ux - ux \cdot ux\right), ux \cdot \left(2 - ux\right)\right) - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0005000000237487257)
   (sqrt
    (-
     (fma -2.0 (* maxCos (- ux (* ux ux))) (* ux (- 2.0 ux)))
     (* (* ux ux) (* maxCos maxCos))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0005000000237487257f) {
		tmp = sqrtf((fmaf(-2.0f, (maxCos * (ux - (ux * ux))), (ux * (2.0f - ux))) - ((ux * ux) * (maxCos * 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(0.0005000000237487257))
		tmp = sqrt(Float32(fma(Float32(-2.0), Float32(maxCos * Float32(ux - Float32(ux * ux))), Float32(ux * Float32(Float32(2.0) - ux))) - Float32(Float32(ux * ux) * Float32(maxCos * maxCos))));
	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 < 5.00000024e-4

   1. Initial program 53.3%

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

      1. associate-*l* 53.3%

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

      2. +-commutative 53.3%

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

      3. associate-+r- 53.3%

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

      4. fma-def 53.3%

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

      5. +-commutative 53.3%

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

      6. associate-+r- 53.2%

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

      7. fma-def 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in ux around -inf 56.8%

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

      1. +-commutative 56.8%

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

      2. mul-1-neg 56.8%

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

      3. unsub-neg 56.8%

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

      4. unpow2 56.8%

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

      5. mul-1-neg 56.8%

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

      6. unsub-neg 56.8%

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

      7. +-commutative 56.8%

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

      8. *-commutative 56.8%

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

      9. fma-def 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in maxCos around 0 99.3%

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

      1. distribute-lft-out-- 99.3%

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

      2. associate-*r* 99.3%

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

      3. metadata-eval 99.3%

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

      4. distribute-rgt-neg-in 99.3%

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

      5. distribute-lft-neg-in 99.3%

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

      6. neg-mul-1 99.3%

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

      7. associate-*r* 99.3%

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

      8. distribute-lft-out-- 99.3%

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

      9. associate-*r* 99.3%

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

      10. associate-+l+ 99.3%

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

      11. +-commutative 99.3%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in uy around 0 97.7%

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

       1. fma-def 97.7%

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

       2. unpow2 97.7%

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

       3. *-commutative 97.7%

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

       4. unpow2 97.7%

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

       5. unpow2 97.7%

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

   12. Simplified 97.7%

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

   if 5.00000024e-4 < uy

   1. Initial program 51.8%

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

      1. associate-*l* 51.8%

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

      2. +-commutative 51.8%

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

      3. associate-+r- 51.9%

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

      4. fma-def 51.9%

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

      5. +-commutative 51.9%

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

      6. associate-+r- 51.8%

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

      7. fma-def 51.8%

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

   3. Simplified 51.8%

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

   4. Taylor expanded in ux around 0 47.0%

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

   5. Taylor expanded in maxCos around 0 78.4%

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

      1. *-commutative 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0005000000237487257:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos \cdot \left(ux - ux \cdot ux\right), ux \cdot \left(2 - ux\right)\right) - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 8: 80.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.9%

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

   4. fma-def 52.9%

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

   5. +-commutative 52.9%

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

   6. associate-+r- 52.8%

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

   7. fma-def 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in ux around -inf 56.8%

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

   1. +-commutative 56.8%

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

   2. mul-1-neg 56.8%

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

   3. unsub-neg 56.8%

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

   4. unpow2 56.8%

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

   5. mul-1-neg 56.8%

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

   6. unsub-neg 56.8%

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

   7. +-commutative 56.8%

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

   8. *-commutative 56.8%

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

   9. fma-def 56.8%

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

6. Simplified 56.8%

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

7. Taylor expanded in maxCos around 0 99.0%

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

   1. distribute-lft-out-- 99.0%

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

   2. associate-*r* 99.0%

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

   3. metadata-eval 99.0%

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

   4. distribute-rgt-neg-in 99.0%

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

   5. distribute-lft-neg-in 99.0%

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

   6. neg-mul-1 99.0%

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

   7. associate-*r* 99.0%

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

   8. distribute-lft-out-- 99.0%

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

   9. associate-*r* 99.0%

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

   10. associate-+l+ 99.0%

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

   11. +-commutative 99.0%

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

9. Simplified 99.0%

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

10. Taylor expanded in uy around 0 81.7%

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

    1. fma-def 81.7%

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

    2. unpow2 81.7%

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

    3. *-commutative 81.7%

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

    4. unpow2 81.7%

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

    5. unpow2 81.7%

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

12. Simplified 81.7%

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

13. Final simplification 81.7%

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

Alternative 9: 80.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.9%

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

   4. fma-def 52.9%

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

   5. +-commutative 52.9%

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

   6. associate-+r- 52.8%

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

   7. fma-def 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in ux around -inf 56.8%

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

   1. +-commutative 56.8%

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

   2. mul-1-neg 56.8%

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

   3. unsub-neg 56.8%

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

   4. unpow2 56.8%

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

   5. mul-1-neg 56.8%

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

   6. unsub-neg 56.8%

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

   7. +-commutative 56.8%

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

   8. *-commutative 56.8%

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

   9. fma-def 56.8%

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

6. Simplified 56.8%

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

7. Taylor expanded in maxCos around 0 99.0%

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

   1. distribute-lft-out-- 99.0%

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

   2. associate-*r* 99.0%

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

   3. metadata-eval 99.0%

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

   4. distribute-rgt-neg-in 99.0%

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

   5. distribute-lft-neg-in 99.0%

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

   6. neg-mul-1 99.0%

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

   7. associate-*r* 99.0%

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

   8. distribute-lft-out-- 99.0%

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

   9. associate-*r* 99.0%

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

   10. associate-+l+ 99.0%

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

   11. +-commutative 99.0%

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

9. Simplified 99.0%

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

10. Taylor expanded in uy around 0 81.7%

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

    1. associate--l+ 81.7%

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

    2. fma-def 81.7%

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

    3. unpow2 81.7%

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

    4. *-commutative 81.7%

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

    5. unpow2 81.7%

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

    6. associate-*l* 81.7%

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

    7. distribute-lft-out-- 81.7%

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

    8. unpow2 81.7%

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

12. Simplified 81.7%

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

13. Final simplification 81.7%

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

Alternative 10: 79.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.9%

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

   4. fma-def 52.9%

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

   5. +-commutative 52.9%

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

   6. associate-+r- 52.8%

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

   7. fma-def 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in ux around -inf 56.8%

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

   1. +-commutative 56.8%

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

   2. mul-1-neg 56.8%

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

   3. unsub-neg 56.8%

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

   4. unpow2 56.8%

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

   5. mul-1-neg 56.8%

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

   6. unsub-neg 56.8%

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

   7. +-commutative 56.8%

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

   8. *-commutative 56.8%

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

   9. fma-def 56.8%

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

6. Simplified 56.8%

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

7. Taylor expanded in maxCos around 0 55.5%

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

8. Taylor expanded in uy around 0 80.5%

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

9. Final simplification 80.5%

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

Alternative 11: 73.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00015999999595806003:\\ \;\;\;\;\sqrt{2 \cdot ux + -2 \cdot \left(maxCos \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - {\left(1 - ux\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00015999999595806003)
   (sqrt (+ (* 2.0 ux) (* -2.0 (* maxCos ux))))
   (sqrt (- 1.0 (pow (- 1.0 ux) 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00015999999595806003f) {
		tmp = sqrtf(((2.0f * ux) + (-2.0f * (maxCos * ux))));
	} else {
		tmp = sqrtf((1.0f - powf((1.0f - ux), 2.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00015999999595806003e0) then
        tmp = sqrt(((2.0e0 * ux) + ((-2.0e0) * (maxcos * ux))))
    else
        tmp = sqrt((1.0e0 - ((1.0e0 - ux) ** 2.0e0)))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00015999999595806003))
		tmp = sqrt(Float32(Float32(Float32(2.0) * ux) + Float32(Float32(-2.0) * Float32(maxCos * ux))));
	else
		tmp = sqrt(Float32(Float32(1.0) - (Float32(Float32(1.0) - ux) ^ Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00015999999595806003))
		tmp = sqrt(((single(2.0) * ux) + (single(-2.0) * (maxCos * ux))));
	else
		tmp = sqrt((single(1.0) - ((single(1.0) - ux) ^ single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.59999996e-4

   1. Initial program 34.0%

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

      1. associate-*l* 34.0%

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

      2. +-commutative 34.0%

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

      3. associate-+r- 34.0%

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

      4. fma-def 34.0%

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

      5. +-commutative 34.0%

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

      6. associate-+r- 34.0%

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

      7. fma-def 34.0%

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

   3. Simplified 34.0%

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

   4. Taylor expanded in uy around 0 30.8%

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

   5. Taylor expanded in ux around 0 77.0%

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

   6. Taylor expanded in maxCos around 0 77.0%

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

   if 1.59999996e-4 < ux

   1. Initial program 88.6%

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

      1. associate-*l* 88.6%

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

      2. +-commutative 88.6%

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

      3. associate-+r- 88.9%

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

      4. fma-def 88.9%

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

      5. +-commutative 88.9%

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

      6. associate-+r- 88.6%

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

      7. fma-def 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in uy around 0 76.8%

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

   5. Taylor expanded in maxCos around 0 72.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00015999999595806003:\\ \;\;\;\;\sqrt{2 \cdot ux + -2 \cdot \left(maxCos \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - {\left(1 - ux\right)}^{2}}\\ \end{array} \]

Alternative 12: 79.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.9%

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

   4. fma-def 52.9%

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

   5. +-commutative 52.9%

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

   6. associate-+r- 52.8%

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

   7. fma-def 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in ux around -inf 56.8%

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

   1. +-commutative 56.8%

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

   2. mul-1-neg 56.8%

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

   3. unsub-neg 56.8%

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

   4. unpow2 56.8%

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

   5. mul-1-neg 56.8%

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

   6. unsub-neg 56.8%

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

   7. +-commutative 56.8%

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

   8. *-commutative 56.8%

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

   9. fma-def 56.8%

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

6. Simplified 56.8%

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

7. Taylor expanded in maxCos around 0 55.5%

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

8. Taylor expanded in uy around 0 80.5%

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

   1. unpow2 80.5%

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

   2. distribute-lft-out-- 80.5%

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

   3. +-commutative 80.5%

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

   4. fma-def 80.5%

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

10. Simplified 80.5%

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

11. Final simplification 80.5%

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

Alternative 13: 64.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.9%

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

   4. fma-def 52.9%

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

   5. +-commutative 52.9%

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

   6. associate-+r- 52.8%

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

   7. fma-def 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in uy around 0 46.6%

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

5. Taylor expanded in ux around 0 67.3%

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

6. Taylor expanded in maxCos around 0 67.4%

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

7. Final simplification 67.4%

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

Alternative 14: 64.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.9%

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

   4. fma-def 52.9%

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

   5. +-commutative 52.9%

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

   6. associate-+r- 52.8%

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

   7. fma-def 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in uy around 0 46.6%

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

5. Taylor expanded in ux around 0 67.3%

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

6. Final simplification 67.3%

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

Alternative 15: 62.0% 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 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.9%

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

   4. fma-def 52.9%

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

   5. +-commutative 52.9%

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

   6. associate-+r- 52.8%

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

   7. fma-def 52.8%

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

3. Simplified 52.8%

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

4. Taylor expanded in uy around 0 46.6%

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

5. Taylor expanded in ux around 0 67.3%

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

6. Taylor expanded in maxCos around 0 63.7%

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

7. Final simplification 63.7%

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

Reproduce

herbie shell --seed 2023272 
(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)))))))