UniformSampleCone, y

Percentage Accurate: 57.2% → 98.4%

Time: 26.8s

Alternatives: 14

Speedup: 1.5×

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

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* 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 sinf(((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(sin(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 = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* 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 sinf(((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(sin(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 = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)}^{1.5}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (cbrt
  (*
   (pow (sin (* 2.0 (* uy PI))) 3.0)
   (pow
    (- (* ux (fma -2.0 maxCos 2.0)) (pow (* ux (+ maxCos -1.0)) 2.0))
    1.5))))

C

float code(float ux, float uy, float maxCos) {
	return cbrtf((powf(sinf((2.0f * (uy * ((float) M_PI)))), 3.0f) * powf(((ux * fmaf(-2.0f, maxCos, 2.0f)) - powf((ux * (maxCos + -1.0f)), 2.0f)), 1.5f)));
}

Julia

function code(ux, uy, maxCos)
	return cbrt(Float32((sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) ^ Float32(3.0)) * (Float32(Float32(ux * fma(Float32(-2.0), maxCos, Float32(2.0))) - (Float32(ux * Float32(maxCos + Float32(-1.0))) ^ Float32(2.0))) ^ Float32(1.5))))
end

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. +-commutative 57.5%

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

   3. associate-+r- 57.5%

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

   4. fma-def 57.5%

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

   5. +-commutative 57.5%

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

   6. associate-+r- 57.4%

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

   7. fma-def 57.4%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. cancel-sign-sub-inv 98.5%

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

   3. metadata-eval 98.5%

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

   4. mul-1-neg 98.5%

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

   5. unsub-neg 98.5%

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

   6. +-commutative 98.5%

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

   7. *-commutative 98.5%

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

   8. fma-def 98.5%

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

   9. unpow2 98.5%

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

   10. associate-*l* 98.5%

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

   11. sub-neg 98.5%

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

   12. metadata-eval 98.5%

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

6. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. add-cbrt-cube 98.5%

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

   3. add-cbrt-cube 98.5%

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

   4. cbrt-unprod 98.5%

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

   5. pow3 98.5%

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

   6. pow3 98.5%

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

8. Applied egg-rr 98.5%

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

10. Applied egg-rr 98.6%

    \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(\pi \cdot \left(uy \cdot 2\right)\right)}^{3} \cdot {\left(\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), -{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)\right)}^{1.5}}} \]
11. Step-by-step derivation

    1. *-commutative 98.6%

       \[\leadsto \sqrt[3]{{\sin \color{blue}{\left(\left(uy \cdot 2\right) \cdot \pi\right)}}^{3} \cdot {\left(\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), -{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)\right)}^{1.5}} \]

    2. *-commutative 98.6%

       \[\leadsto \sqrt[3]{{\sin \left(\color{blue}{\left(2 \cdot uy\right)} \cdot \pi\right)}^{3} \cdot {\left(\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), -{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)\right)}^{1.5}} \]

    3. associate-*r* 98.6%

       \[\leadsto \sqrt[3]{{\sin \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)}}^{3} \cdot {\left(\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), -{\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)\right)}^{1.5}} \]

12. Simplified 98.6%

    \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)}^{1.5}}} \]

13. Final simplification 98.6%

    \[\leadsto \sqrt[3]{{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)}^{1.5}} \]

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)}^{1.5}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* uy (* 2.0 PI)))
  (cbrt
   (pow
    (* ux (- (fma -2.0 maxCos 2.0) (* ux (pow (+ maxCos -1.0) 2.0))))
    1.5))))

C

float code(float ux, float uy, float maxCos) {
	return sinf((uy * (2.0f * ((float) M_PI)))) * cbrtf(powf((ux * (fmaf(-2.0f, maxCos, 2.0f) - (ux * powf((maxCos + -1.0f), 2.0f)))), 1.5f));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * cbrt((Float32(ux * Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) - Float32(ux * (Float32(maxCos + Float32(-1.0)) ^ Float32(2.0))))) ^ Float32(1.5))))
end

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. +-commutative 57.5%

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

   3. associate-+r- 57.5%

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

   4. fma-def 57.5%

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

   5. +-commutative 57.5%

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

   6. associate-+r- 57.4%

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

   7. fma-def 57.4%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. cancel-sign-sub-inv 98.5%

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

   3. metadata-eval 98.5%

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

   4. mul-1-neg 98.5%

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

   5. unsub-neg 98.5%

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

   6. +-commutative 98.5%

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

   7. *-commutative 98.5%

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

   8. fma-def 98.5%

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

   9. unpow2 98.5%

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

   10. associate-*l* 98.5%

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

   11. sub-neg 98.5%

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

   12. metadata-eval 98.5%

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

6. Simplified 98.5%

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

   1. add-cbrt-cube 98.4%

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

   2. add-sqr-sqrt 98.4%

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

   3. distribute-lft-out-- 98.4%

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

   4. distribute-lft-out-- 98.4%

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

8. Applied egg-rr 98.4%

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\sqrt[3]{\left(ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right) \cdot \sqrt{ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)}}} \]
9. Step-by-step derivation

   1. *-commutative 98.4%

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

   2. unpow1/2 98.4%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\color{blue}{{\left(ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)}^{0.5}} \cdot \left(ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)} \]

   3. pow-plus 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{\color{blue}{{\left(ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)}^{\left(0.5 + 1\right)}}} \]

   4. fma-udef 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\color{blue}{\left(maxCos \cdot -2 + 2\right)} - ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)}^{\left(0.5 + 1\right)}} \]

   5. *-commutative 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\left(\color{blue}{-2 \cdot maxCos} + 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)}^{\left(0.5 + 1\right)}} \]

   6. fma-def 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} - ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)}^{\left(0.5 + 1\right)}} \]

   7. metadata-eval 98.5%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)}^{\color{blue}{1.5}}} \]

10. Simplified 98.5%

    \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)}^{1.5}}} \]

11. Final simplification 98.5%

    \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt[3]{{\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)}^{1.5}} \]

Alternative 3: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. +-commutative 57.5%

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

   3. associate-+r- 57.5%

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

   4. fma-def 57.5%

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

   5. +-commutative 57.5%

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

   6. associate-+r- 57.4%

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

   7. fma-def 57.4%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. cancel-sign-sub-inv 98.5%

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

   3. metadata-eval 98.5%

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

   4. mul-1-neg 98.5%

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

   5. unsub-neg 98.5%

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

   6. +-commutative 98.5%

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

   7. *-commutative 98.5%

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

   8. fma-def 98.5%

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

   9. unpow2 98.5%

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

   10. associate-*l* 98.5%

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

   11. sub-neg 98.5%

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

   12. metadata-eval 98.5%

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

6. Simplified 98.5%

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

   1. pow1 98.5%

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

   2. *-commutative 98.5%

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

   3. distribute-lft-out-- 98.5%

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

8. Applied egg-rr 98.5%

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

   1. unpow1 98.5%

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

   2. *-commutative 98.5%

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

10. Simplified 98.5%

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

11. Final simplification 98.5%

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

Alternative 4: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. +-commutative 57.5%

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

   3. associate-+r- 57.5%

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

   4. fma-def 57.5%

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

   5. +-commutative 57.5%

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

   6. associate-+r- 57.4%

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

   7. fma-def 57.4%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. cancel-sign-sub-inv 98.5%

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

   3. metadata-eval 98.5%

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

   4. mul-1-neg 98.5%

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

   5. unsub-neg 98.5%

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

   6. +-commutative 98.5%

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

   7. *-commutative 98.5%

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

   8. fma-def 98.5%

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

   9. unpow2 98.5%

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

   10. associate-*l* 98.5%

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

   11. sub-neg 98.5%

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

   12. metadata-eval 98.5%

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

6. Simplified 98.5%

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

   1. associate-*r* 98.5%

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

   2. add-cbrt-cube 98.5%

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

   3. add-cbrt-cube 98.5%

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

   4. cbrt-unprod 98.5%

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

   5. pow3 98.5%

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

   6. pow3 98.5%

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

8. Applied egg-rr 98.5%

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

9. Taylor expanded in uy around inf 98.5%

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

    1. *-commutative 98.5%

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

    2. unpow2 98.5%

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

    3. unpow2 98.5%

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

    4. swap-sqr 98.5%

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

    5. sub-neg 98.5%

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

    6. metadata-eval 98.5%

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

    7. sub-neg 98.5%

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

    8. metadata-eval 98.5%

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

    9. unpow2 98.5%

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

    10. +-commutative 98.5%

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

    11. fma-def 98.5%

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

11. Simplified 98.5%

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

12. Final simplification 98.5%

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

Alternative 5: 95.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot ux - ux \cdot ux\\ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{t_0} + \left(0.5 \cdot \left(maxCos \cdot \left(-2 \cdot \left(ux - ux \cdot ux\right)\right)\right)\right) \cdot \sqrt{\frac{1}{t_0}}\right) \end{array} \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (2.0f * ux) - (ux * ux);
	return sinf((uy * (2.0f * ((float) M_PI)))) * (sqrtf(t_0) + ((0.5f * (maxCos * (-2.0f * (ux - (ux * ux))))) * sqrtf((1.0f / t_0))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))
	return Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * Float32(sqrt(t_0) + Float32(Float32(Float32(0.5) * Float32(maxCos * Float32(Float32(-2.0) * Float32(ux - Float32(ux * ux))))) * sqrt(Float32(Float32(1.0) / t_0)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(2.0) * ux) - (ux * ux);
	tmp = sin((uy * (single(2.0) * single(pi)))) * (sqrt(t_0) + ((single(0.5) * (maxCos * (single(-2.0) * (ux - (ux * ux))))) * sqrt((single(1.0) / t_0))));
end

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. +-commutative 57.5%

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

   3. associate-+r- 57.5%

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

   4. fma-def 57.5%

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

   5. +-commutative 57.5%

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

   6. associate-+r- 57.4%

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

   7. fma-def 57.4%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. cancel-sign-sub-inv 98.5%

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

   3. metadata-eval 98.5%

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

   4. mul-1-neg 98.5%

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

   5. unsub-neg 98.5%

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

   6. +-commutative 98.5%

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

   7. *-commutative 98.5%

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

   8. fma-def 98.5%

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

   9. unpow2 98.5%

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

   10. associate-*l* 98.5%

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

   11. sub-neg 98.5%

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

   12. metadata-eval 98.5%

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

6. Simplified 98.5%

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

7. Taylor expanded in maxCos around 0 96.0%

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\left(\sqrt{2 \cdot ux - {ux}^{2}} + 0.5 \cdot \left(\left(maxCos \cdot \left(-2 \cdot ux - -2 \cdot {ux}^{2}\right)\right) \cdot \sqrt{\frac{1}{2 \cdot ux - {ux}^{2}}}\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 96.0%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{2 \cdot ux - \color{blue}{ux \cdot ux}} + 0.5 \cdot \left(\left(maxCos \cdot \left(-2 \cdot ux - -2 \cdot {ux}^{2}\right)\right) \cdot \sqrt{\frac{1}{2 \cdot ux - {ux}^{2}}}\right)\right) \]

   2. associate-*r* 96.0%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{2 \cdot ux - ux \cdot ux} + \color{blue}{\left(0.5 \cdot \left(maxCos \cdot \left(-2 \cdot ux - -2 \cdot {ux}^{2}\right)\right)\right) \cdot \sqrt{\frac{1}{2 \cdot ux - {ux}^{2}}}}\right) \]

   3. distribute-lft-out-- 96.0%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{2 \cdot ux - ux \cdot ux} + \left(0.5 \cdot \left(maxCos \cdot \color{blue}{\left(-2 \cdot \left(ux - {ux}^{2}\right)\right)}\right)\right) \cdot \sqrt{\frac{1}{2 \cdot ux - {ux}^{2}}}\right) \]

   4. unpow2 96.0%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{2 \cdot ux - ux \cdot ux} + \left(0.5 \cdot \left(maxCos \cdot \left(-2 \cdot \left(ux - \color{blue}{ux \cdot ux}\right)\right)\right)\right) \cdot \sqrt{\frac{1}{2 \cdot ux - {ux}^{2}}}\right) \]

   5. unpow2 96.0%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{2 \cdot ux - ux \cdot ux} + \left(0.5 \cdot \left(maxCos \cdot \left(-2 \cdot \left(ux - ux \cdot ux\right)\right)\right)\right) \cdot \sqrt{\frac{1}{2 \cdot ux - \color{blue}{ux \cdot ux}}}\right) \]

9. Simplified 96.0%

   \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\left(\sqrt{2 \cdot ux - ux \cdot ux} + \left(0.5 \cdot \left(maxCos \cdot \left(-2 \cdot \left(ux - ux \cdot ux\right)\right)\right)\right) \cdot \sqrt{\frac{1}{2 \cdot ux - ux \cdot ux}}\right)} \]

10. Final simplification 96.0%

    \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\sqrt{2 \cdot ux - ux \cdot ux} + \left(0.5 \cdot \left(maxCos \cdot \left(-2 \cdot \left(ux - ux \cdot ux\right)\right)\right)\right) \cdot \sqrt{\frac{1}{2 \cdot ux - ux \cdot ux}}\right) \]

Alternative 6: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00022000000171829015:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.00022000000171829015)
   (*
    2.0
    (*
     uy
     (*
      PI
      (sqrt
       (- (* ux (fma -2.0 maxCos 2.0)) (pow (* ux (+ maxCos -1.0)) 2.0))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (- (* 2.0 ux) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.00022000000171829015f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((ux * fmaf(-2.0f, maxCos, 2.0f)) - powf((ux * (maxCos + -1.0f)), 2.0f)))));
	} else {
		tmp = sinf((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 (Float32(Float32(2.0) * uy) <= Float32(0.00022000000171829015))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(ux * fma(Float32(-2.0), maxCos, Float32(2.0))) - (Float32(ux * Float32(maxCos + Float32(-1.0))) ^ Float32(2.0)))))));
	else
		tmp = Float32(sin(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 (*.f32 uy 2) < 2.20000002e-4

   1. Initial program 57.7%

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

      1. associate-*l* 57.7%

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

      2. +-commutative 57.7%

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

      3. associate-+r- 57.8%

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

      4. fma-def 57.8%

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

      5. +-commutative 57.8%

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

      6. associate-+r- 57.7%

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

      7. fma-def 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in ux around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. metadata-eval 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. +-commutative 98.7%

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

      7. *-commutative 98.7%

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

      8. fma-def 98.7%

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

      9. unpow2 98.7%

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

      10. associate-*l* 98.7%

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

      11. sub-neg 98.7%

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

      12. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

      1. associate-*r* 98.7%

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

      2. add-cbrt-cube 98.7%

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

      3. add-cbrt-cube 98.7%

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

      4. cbrt-unprod 98.6%

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

      5. pow3 98.7%

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

      6. pow3 98.7%

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

   8. Applied egg-rr 98.7%

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

   9. Taylor expanded in uy around 0 98.7%

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

       1. associate-*l* 98.4%

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

       2. unpow2 98.4%

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

       3. unpow2 98.4%

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

       4. swap-sqr 98.4%

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

       5. sub-neg 98.4%

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

       6. metadata-eval 98.4%

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

       7. sub-neg 98.4%

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

       8. metadata-eval 98.4%

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

       9. unpow2 98.4%

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

       10. +-commutative 98.4%

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

       11. fma-def 98.4%

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

   11. Simplified 98.4%

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

   if 2.20000002e-4 < (*.f32 uy 2)

   1. Initial program 57.1%

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

      1. associate-*l* 57.1%

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

      2. +-commutative 57.1%

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

      3. associate-+r- 57.2%

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

      4. fma-def 57.2%

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

      5. +-commutative 57.2%

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

      6. associate-+r- 57.0%

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

      7. fma-def 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in ux around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. cancel-sign-sub-inv 98.2%

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

      3. metadata-eval 98.2%

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

      4. mul-1-neg 98.2%

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

      5. unsub-neg 98.2%

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

      6. +-commutative 98.2%

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

      7. *-commutative 98.2%

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

      8. fma-def 98.2%

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

      9. unpow2 98.2%

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

      10. associate-*l* 98.2%

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

      11. sub-neg 98.2%

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

      12. metadata-eval 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in maxCos around 0 92.4%

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

      1. *-commutative 92.4%

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

      2. associate-*r* 92.4%

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

      3. *-commutative 92.4%

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

      4. unpow2 92.4%

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

   9. Simplified 92.4%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00022000000171829015:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \]

Alternative 7: 95.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;2 \cdot uy \leq 0.00022000000171829015:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (if (<= (* 2.0 uy) 0.00022000000171829015)
     (*
      t_0
      (sqrt (* ux (- (fma maxCos -2.0 2.0) (* ux (pow (+ maxCos -1.0) 2.0))))))
     (* (sin t_0) (sqrt (- (* 2.0 ux) (* ux ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = uy * (2.0f * ((float) M_PI));
	float tmp;
	if ((2.0f * uy) <= 0.00022000000171829015f) {
		tmp = t_0 * sqrtf((ux * (fmaf(maxCos, -2.0f, 2.0f) - (ux * powf((maxCos + -1.0f), 2.0f)))));
	} else {
		tmp = sinf(t_0) * sqrtf(((2.0f * ux) - (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.00022000000171829015))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) - Float32(ux * (Float32(maxCos + Float32(-1.0)) ^ Float32(2.0)))))));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 2.20000002e-4

   1. Initial program 57.7%

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

      1. associate-*l* 57.7%

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

      2. +-commutative 57.7%

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

      3. associate-+r- 57.8%

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

      4. fma-def 57.8%

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

      5. +-commutative 57.8%

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

      6. associate-+r- 57.7%

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

      7. fma-def 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in ux around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. metadata-eval 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. +-commutative 98.7%

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

      7. *-commutative 98.7%

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

      8. fma-def 98.7%

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

      9. unpow2 98.7%

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

      10. associate-*l* 98.7%

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

      11. sub-neg 98.7%

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

      12. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in uy around 0 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*r* 98.7%

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

      3. *-commutative 98.7%

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

   9. Simplified 98.7%

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

       1. *-un-lft-identity 98.7%

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

       2. distribute-lft-out-- 98.7%

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

   11. Applied egg-rr 98.7%

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

       1. *-lft-identity 98.7%

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

   13. Simplified 98.7%

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

   if 2.20000002e-4 < (*.f32 uy 2)

   1. Initial program 57.1%

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

      1. associate-*l* 57.1%

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

      2. +-commutative 57.1%

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

      3. associate-+r- 57.2%

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

      4. fma-def 57.2%

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

      5. +-commutative 57.2%

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

      6. associate-+r- 57.0%

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

      7. fma-def 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in ux around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. cancel-sign-sub-inv 98.2%

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

      3. metadata-eval 98.2%

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

      4. mul-1-neg 98.2%

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

      5. unsub-neg 98.2%

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

      6. +-commutative 98.2%

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

      7. *-commutative 98.2%

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

      8. fma-def 98.2%

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

      9. unpow2 98.2%

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

      10. associate-*l* 98.2%

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

      11. sub-neg 98.2%

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

      12. metadata-eval 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in maxCos around 0 92.4%

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

      1. *-commutative 92.4%

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

      2. associate-*r* 92.4%

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

      3. *-commutative 92.4%

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

      4. unpow2 92.4%

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

   9. Simplified 92.4%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00022000000171829015:\\ \;\;\;\;\left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \]

Alternative 8: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;2 \cdot uy \leq 0.00022000000171829015:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) + \left(ux \cdot ux\right) \cdot \left(-1 - -2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (if (<= (* 2.0 uy) 0.00022000000171829015)
     (*
      t_0
      (sqrt
       (+
        (* ux (fma maxCos -2.0 2.0))
        (* (* ux ux) (- -1.0 (* -2.0 maxCos))))))
     (* (sin t_0) (sqrt (- (* 2.0 ux) (* ux ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = uy * (2.0f * ((float) M_PI));
	float tmp;
	if ((2.0f * uy) <= 0.00022000000171829015f) {
		tmp = t_0 * sqrtf(((ux * fmaf(maxCos, -2.0f, 2.0f)) + ((ux * ux) * (-1.0f - (-2.0f * maxCos)))));
	} else {
		tmp = sinf(t_0) * sqrtf(((2.0f * ux) - (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.00022000000171829015))
		tmp = Float32(t_0 * sqrt(Float32(Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0))) + Float32(Float32(ux * ux) * Float32(Float32(-1.0) - Float32(Float32(-2.0) * maxCos))))));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 2.20000002e-4

   1. Initial program 57.7%

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

      1. associate-*l* 57.7%

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

      2. +-commutative 57.7%

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

      3. associate-+r- 57.8%

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

      4. fma-def 57.8%

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

      5. +-commutative 57.8%

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

      6. associate-+r- 57.7%

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

      7. fma-def 57.7%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in ux around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. metadata-eval 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. +-commutative 98.7%

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

      7. *-commutative 98.7%

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

      8. fma-def 98.7%

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

      9. unpow2 98.7%

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

      10. associate-*l* 98.7%

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

      11. sub-neg 98.7%

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

      12. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in uy around 0 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*r* 98.7%

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

      3. *-commutative 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in maxCos around 0 98.3%

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

       1. associate-*r* 98.3%

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

       2. distribute-lft1-in 98.3%

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

       3. unpow2 98.3%

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

   12. Simplified 98.3%

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

   if 2.20000002e-4 < (*.f32 uy 2)

   1. Initial program 57.1%

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

      1. associate-*l* 57.1%

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

      2. +-commutative 57.1%

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

      3. associate-+r- 57.2%

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

      4. fma-def 57.2%

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

      5. +-commutative 57.2%

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

      6. associate-+r- 57.0%

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

      7. fma-def 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in ux around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. cancel-sign-sub-inv 98.2%

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

      3. metadata-eval 98.2%

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

      4. mul-1-neg 98.2%

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

      5. unsub-neg 98.2%

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

      6. +-commutative 98.2%

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

      7. *-commutative 98.2%

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

      8. fma-def 98.2%

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

      9. unpow2 98.2%

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

      10. associate-*l* 98.2%

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

      11. sub-neg 98.2%

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

      12. metadata-eval 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in maxCos around 0 92.4%

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

      1. *-commutative 92.4%

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

      2. associate-*r* 92.4%

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

      3. *-commutative 92.4%

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

      4. unpow2 92.4%

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

   9. Simplified 92.4%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00022000000171829015:\\ \;\;\;\;\left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) + \left(ux \cdot ux\right) \cdot \left(-1 - -2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \]

Alternative 9: 92.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. +-commutative 57.5%

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

   3. associate-+r- 57.5%

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

   4. fma-def 57.5%

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

   5. +-commutative 57.5%

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

   6. associate-+r- 57.4%

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

   7. fma-def 57.4%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. cancel-sign-sub-inv 98.5%

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

   3. metadata-eval 98.5%

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

   4. mul-1-neg 98.5%

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

   5. unsub-neg 98.5%

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

   6. +-commutative 98.5%

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

   7. *-commutative 98.5%

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

   8. fma-def 98.5%

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

   9. unpow2 98.5%

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

   10. associate-*l* 98.5%

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

   11. sub-neg 98.5%

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

   12. metadata-eval 98.5%

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

6. Simplified 98.5%

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

7. Taylor expanded in maxCos around 0 92.6%

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

   1. *-commutative 92.6%

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

   2. associate-*r* 92.6%

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

   3. *-commutative 92.6%

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

   4. unpow2 92.6%

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

9. Simplified 92.6%

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

10. Final simplification 92.6%

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

Alternative 10: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0002300000051036477:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - ux\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0002300000051036477)
   (* 2.0 (* uy (* PI (sqrt (* ux (- 2.0 ux))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0002300000051036477f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * (2.0f - ux)))));
	} else {
		tmp = sinf((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(0.0002300000051036477))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0002300000051036477))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * (single(2.0) - ux)))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 2.30000005e-4

   1. Initial program 58.8%

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

      1. associate-*l* 58.8%

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

      2. +-commutative 58.8%

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

      3. associate-+r- 58.8%

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

      4. fma-def 58.8%

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

      5. +-commutative 58.8%

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

      6. associate-+r- 58.7%

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

      7. fma-def 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in ux around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. metadata-eval 98.7%

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

      4. mul-1-neg 98.7%

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

      5. unsub-neg 98.7%

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

      6. +-commutative 98.7%

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

      7. *-commutative 98.7%

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

      8. fma-def 98.7%

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

      9. unpow2 98.7%

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

      10. associate-*l* 98.7%

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

      11. sub-neg 98.7%

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

      12. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in uy around 0 98.4%

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

      1. *-commutative 98.4%

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

      2. associate-*r* 98.4%

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

      3. *-commutative 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in maxCos around 0 92.8%

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

       1. associate-*l* 92.7%

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

       2. unpow2 92.7%

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

       3. distribute-rgt-out-- 92.8%

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

   12. Simplified 92.8%

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

   if 2.30000005e-4 < uy

   1. Initial program 55.5%

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

      1. associate-*l* 55.5%

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

      2. +-commutative 55.5%

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

      3. associate-+r- 55.5%

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

      4. fma-def 55.5%

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

      5. +-commutative 55.5%

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

      6. associate-+r- 55.3%

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

      7. fma-def 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in ux around 0 46.3%

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

   5. Taylor expanded in maxCos around 0 75.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0002300000051036477:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - ux\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 11: 92.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. +-commutative 57.5%

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

   3. associate-+r- 57.5%

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

   4. fma-def 57.5%

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

   5. +-commutative 57.5%

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

   6. associate-+r- 57.4%

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

   7. fma-def 57.4%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. cancel-sign-sub-inv 98.5%

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

   3. metadata-eval 98.5%

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

   4. mul-1-neg 98.5%

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

   5. unsub-neg 98.5%

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

   6. +-commutative 98.5%

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

   7. *-commutative 98.5%

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

   8. fma-def 98.5%

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

   9. unpow2 98.5%

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

   10. associate-*l* 98.5%

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

   11. sub-neg 98.5%

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

   12. metadata-eval 98.5%

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

6. Simplified 98.5%

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

7. Taylor expanded in maxCos around 0 92.6%

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

   1. unpow2 76.6%

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

   2. distribute-rgt-out-- 76.6%

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

9. Simplified 92.6%

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

10. Final simplification 92.6%

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

Alternative 12: 77.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. +-commutative 57.5%

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

   3. associate-+r- 57.5%

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

   4. fma-def 57.5%

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

   5. +-commutative 57.5%

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

   6. associate-+r- 57.4%

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

   7. fma-def 57.4%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. cancel-sign-sub-inv 98.5%

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

   3. metadata-eval 98.5%

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

   4. mul-1-neg 98.5%

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

   5. unsub-neg 98.5%

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

   6. +-commutative 98.5%

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

   7. *-commutative 98.5%

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

   8. fma-def 98.5%

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

   9. unpow2 98.5%

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

   10. associate-*l* 98.5%

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

   11. sub-neg 98.5%

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

   12. metadata-eval 98.5%

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

6. Simplified 98.5%

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

7. Taylor expanded in uy around 0 80.4%

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

   1. *-commutative 80.4%

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

   2. associate-*r* 80.4%

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

   3. *-commutative 80.4%

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

9. Simplified 80.4%

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

10. Taylor expanded in maxCos around 0 76.6%

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

    1. associate-*l* 76.5%

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

    2. unpow2 76.5%

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

    3. distribute-rgt-out-- 76.6%

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

12. Simplified 76.6%

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

13. Final simplification 76.6%

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

Alternative 13: 77.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. +-commutative 57.5%

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

   3. associate-+r- 57.5%

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

   4. fma-def 57.5%

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

   5. +-commutative 57.5%

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

   6. associate-+r- 57.4%

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

   7. fma-def 57.4%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. cancel-sign-sub-inv 98.5%

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

   3. metadata-eval 98.5%

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

   4. mul-1-neg 98.5%

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

   5. unsub-neg 98.5%

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

   6. +-commutative 98.5%

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

   7. *-commutative 98.5%

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

   8. fma-def 98.5%

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

   9. unpow2 98.5%

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

   10. associate-*l* 98.5%

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

   11. sub-neg 98.5%

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

   12. metadata-eval 98.5%

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

6. Simplified 98.5%

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

7. Taylor expanded in uy around 0 80.4%

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

   1. *-commutative 80.4%

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

   2. associate-*r* 80.4%

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

   3. *-commutative 80.4%

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

9. Simplified 80.4%

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

10. Taylor expanded in maxCos around 0 76.6%

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

    1. unpow2 76.6%

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

    2. distribute-rgt-out-- 76.6%

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

12. Simplified 76.6%

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

13. Final simplification 76.6%

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

Alternative 14: 63.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

   1. associate-*l* 57.5%

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

   2. +-commutative 57.5%

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

   3. associate-+r- 57.5%

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

   4. fma-def 57.5%

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

   5. +-commutative 57.5%

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

   6. associate-+r- 57.4%

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

   7. fma-def 57.4%

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

3. Simplified 57.4%

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

4. Taylor expanded in ux around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. cancel-sign-sub-inv 98.5%

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

   3. metadata-eval 98.5%

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

   4. mul-1-neg 98.5%

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

   5. unsub-neg 98.5%

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

   6. +-commutative 98.5%

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

   7. *-commutative 98.5%

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

   8. fma-def 98.5%

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

   9. unpow2 98.5%

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

   10. associate-*l* 98.5%

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

   11. sub-neg 98.5%

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

   12. metadata-eval 98.5%

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

6. Simplified 98.5%

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

7. Taylor expanded in uy around 0 80.4%

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

   1. *-commutative 80.4%

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

   2. associate-*r* 80.4%

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

   3. *-commutative 80.4%

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

9. Simplified 80.4%

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

10. Taylor expanded in maxCos around 0 76.6%

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

    1. associate-*l* 76.5%

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

    2. unpow2 76.5%

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

    3. distribute-rgt-out-- 76.6%

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

12. Simplified 76.6%

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

13. Taylor expanded in ux around 0 64.0%

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

14. Final simplification 64.0%

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

Reproduce

herbie shell --seed 2023275 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :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)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))