UniformSampleCone, y

Percentage Accurate: 57.3% → 98.3%

Time: 19.6s

Alternatives: 8

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: 57.3% 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: 98.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.8%

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

   4. fma-def 52.8%

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

   5. +-commutative 52.8%

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

   6. associate-+r- 52.7%

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

   7. fma-def 52.7%

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

3. Simplified 52.7%

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

4. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. metadata-eval 98.3%

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

   3. cancel-sign-sub-inv 98.3%

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

   4. fma-def 98.4%

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

   5. cancel-sign-sub-inv 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

   9. fma-def 98.4%

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

   10. mul-1-neg 98.4%

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

   11. *-commutative 98.4%

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

   12. distribute-rgt-neg-in 98.4%

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

   13. mul-1-neg 98.4%

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

   14. unsub-neg 98.4%

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

   15. unpow2 98.4%

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

   16. distribute-rgt-neg-in 98.4%

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

6. Simplified 98.4%

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

   1. associate-*r* 98.4%

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

   2. add-cbrt-cube 98.4%

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

   3. add-cbrt-cube 98.4%

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

   4. cbrt-unprod 98.5%

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

   5. pow3 98.5%

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

   6. pow3 98.5%

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.8%

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

   4. fma-def 52.8%

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

   5. +-commutative 52.8%

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

   6. associate-+r- 52.7%

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

   7. fma-def 52.7%

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

3. Simplified 52.7%

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

4. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. metadata-eval 98.3%

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

   3. cancel-sign-sub-inv 98.3%

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

   4. fma-def 98.4%

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

   5. cancel-sign-sub-inv 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

   9. fma-def 98.4%

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

   10. mul-1-neg 98.4%

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

   11. *-commutative 98.4%

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

   12. distribute-rgt-neg-in 98.4%

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

   13. mul-1-neg 98.4%

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

   14. unsub-neg 98.4%

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

   15. unpow2 98.4%

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

   16. distribute-rgt-neg-in 98.4%

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

6. Simplified 98.4%

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

   1. associate-*r* 98.4%

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

   2. add-cbrt-cube 98.4%

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

   3. add-cbrt-cube 98.4%

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

   4. cbrt-unprod 98.5%

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

   5. pow3 98.5%

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

   6. pow3 98.5%

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

8. Applied egg-rr 98.5%

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

9. Taylor expanded in uy around inf 98.3%

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

    1. *-commutative 98.3%

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

11. Simplified 98.3%

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

    1. *-un-lft-identity 98.3%

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

    2. *-commutative 98.3%

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

13. Applied egg-rr 98.3%

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

    1. *-lft-identity 98.3%

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

    2. distribute-lft-out-- 98.4%

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

15. Simplified 98.4%

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

16. Final simplification 98.4%

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

Alternative 3: 95.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.8%

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

   4. fma-def 52.8%

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

   5. +-commutative 52.8%

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

   6. associate-+r- 52.7%

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

   7. fma-def 52.7%

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

3. Simplified 52.7%

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

4. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. metadata-eval 98.3%

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

   3. cancel-sign-sub-inv 98.3%

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

   4. fma-def 98.4%

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

   5. cancel-sign-sub-inv 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

   9. fma-def 98.4%

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

   10. mul-1-neg 98.4%

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

   11. *-commutative 98.4%

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

   12. distribute-rgt-neg-in 98.4%

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

   13. mul-1-neg 98.4%

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

   14. unsub-neg 98.4%

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

   15. unpow2 98.4%

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

   16. distribute-rgt-neg-in 98.4%

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

6. Simplified 98.4%

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

   1. associate-*r* 98.4%

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

   2. add-cbrt-cube 98.4%

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

   3. add-cbrt-cube 98.4%

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

   4. cbrt-unprod 98.5%

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

   5. pow3 98.5%

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

   6. pow3 98.5%

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

8. Applied egg-rr 98.5%

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

9. Taylor expanded in uy around inf 98.3%

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

    1. *-commutative 98.3%

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

11. Simplified 98.3%

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

12. Taylor expanded in maxCos around 0 96.0%

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

    1. +-commutative 96.0%

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

    2. *-commutative 96.0%

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

    3. unpow2 96.0%

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

    4. distribute-lft-out-- 96.0%

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

    5. unpow2 96.0%

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

    6. unpow2 96.0%

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

14. Simplified 96.0%

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

15. Final simplification 96.0%

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

Alternative 4: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if maxCos < 3.99999999e-6

   1. Initial program 53.9%

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

      1. associate-*l* 53.9%

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

      2. +-commutative 53.9%

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

      3. associate-+r- 53.9%

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

      4. fma-def 53.9%

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

      5. +-commutative 53.9%

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

      6. associate-+r- 53.9%

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

      7. fma-def 53.9%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in ux around -inf 98.4%

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

      1. +-commutative 98.4%

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

      2. metadata-eval 98.4%

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

      3. cancel-sign-sub-inv 98.4%

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

      4. fma-def 98.4%

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

      5. cancel-sign-sub-inv 98.4%

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

      6. metadata-eval 98.4%

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

      7. +-commutative 98.4%

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

      8. *-commutative 98.4%

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

      9. fma-def 98.4%

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

      10. mul-1-neg 98.4%

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

      11. *-commutative 98.4%

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

      12. distribute-rgt-neg-in 98.4%

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

      13. mul-1-neg 98.4%

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

      14. unsub-neg 98.4%

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

      15. unpow2 98.4%

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

      16. distribute-rgt-neg-in 98.4%

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

   6. Simplified 98.4%

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

      1. associate-*r* 98.4%

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

      2. add-cbrt-cube 98.4%

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

      3. add-cbrt-cube 98.4%

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

      4. cbrt-unprod 98.5%

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

      5. pow3 98.6%

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

      6. pow3 98.6%

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

   8. Applied egg-rr 98.6%

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

   9. Taylor expanded in maxCos around 0 98.1%

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

       1. *-commutative 98.1%

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

       2. associate-*r* 98.1%

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

       3. *-commutative 98.1%

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

       4. associate-*l* 98.1%

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

       5. +-commutative 98.1%

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

       6. *-commutative 98.1%

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

       7. mul-1-neg 98.1%

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

       8. unpow2 98.1%

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

       9. distribute-rgt-neg-out 98.1%

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

       10. distribute-lft-out 98.1%

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

       11. unsub-neg 98.1%

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

   11. Simplified 98.1%

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

   if 3.99999999e-6 < maxCos

   1. Initial program 47.2%

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

      1. associate-*l* 47.2%

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

      2. +-commutative 47.2%

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

      3. associate-+r- 46.7%

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

      4. fma-def 46.7%

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

      5. +-commutative 46.7%

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

      6. associate-+r- 46.3%

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

      7. fma-def 46.3%

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

   3. Simplified 46.3%

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

   4. Taylor expanded in ux around 0 81.5%

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

4. Final simplification 95.5%

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

Alternative 5: 92.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.8%

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

   4. fma-def 52.8%

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

   5. +-commutative 52.8%

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

   6. associate-+r- 52.7%

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

   7. fma-def 52.7%

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

3. Simplified 52.7%

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

4. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. metadata-eval 98.3%

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

   3. cancel-sign-sub-inv 98.3%

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

   4. fma-def 98.4%

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

   5. cancel-sign-sub-inv 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

   9. fma-def 98.4%

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

   10. mul-1-neg 98.4%

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

   11. *-commutative 98.4%

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

   12. distribute-rgt-neg-in 98.4%

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

   13. mul-1-neg 98.4%

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

   14. unsub-neg 98.4%

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

   15. unpow2 98.4%

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

   16. distribute-rgt-neg-in 98.4%

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

6. Simplified 98.4%

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

   1. associate-*r* 98.4%

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

   2. add-cbrt-cube 98.4%

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

   3. add-cbrt-cube 98.4%

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

   4. cbrt-unprod 98.5%

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

   5. pow3 98.5%

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

   6. pow3 98.5%

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

8. Applied egg-rr 98.5%

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

9. Taylor expanded in maxCos around 0 92.0%

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

    1. *-commutative 92.0%

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

    2. associate-*r* 92.0%

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

    3. *-commutative 92.0%

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

    4. associate-*l* 92.0%

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

    5. +-commutative 92.0%

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

    6. *-commutative 92.0%

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

    7. mul-1-neg 92.0%

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

    8. unpow2 92.0%

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

    9. distribute-rgt-neg-out 92.0%

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

    10. distribute-lft-out 92.1%

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

    11. unsub-neg 92.1%

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

11. Simplified 92.1%

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

12. Final simplification 92.1%

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

Alternative 6: 77.1% accurate, 1.5× speedup

Math

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

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(uy * Float32(Float32(2.0) * Float32(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 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.8%

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

   4. fma-def 52.8%

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

   5. +-commutative 52.8%

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

   6. associate-+r- 52.7%

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

   7. fma-def 52.7%

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

3. Simplified 52.7%

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

4. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. metadata-eval 98.3%

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

   3. cancel-sign-sub-inv 98.3%

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

   4. fma-def 98.4%

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

   5. cancel-sign-sub-inv 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

   9. fma-def 98.4%

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

   10. mul-1-neg 98.4%

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

   11. *-commutative 98.4%

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

   12. distribute-rgt-neg-in 98.4%

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

   13. mul-1-neg 98.4%

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

   14. unsub-neg 98.4%

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

   15. unpow2 98.4%

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

   16. distribute-rgt-neg-in 98.4%

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

6. Simplified 98.4%

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

7. Taylor expanded in maxCos around 0 92.0%

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

   1. fma-def 92.0%

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

   2. unpow2 92.0%

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

9. Simplified 92.0%

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

10. Taylor expanded in uy around 0 74.4%

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

    1. *-commutative 74.4%

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

    2. associate-*l* 74.4%

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

    3. associate-*l* 74.4%

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

    4. +-commutative 74.4%

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

    5. *-commutative 74.4%

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

    6. mul-1-neg 74.4%

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

    7. unpow2 74.4%

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

    8. distribute-rgt-neg-out 74.4%

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

    9. distribute-lft-out 74.4%

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

    10. unsub-neg 74.4%

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

12. Simplified 74.4%

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

13. Final simplification 74.4%

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

Alternative 7: 77.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.8%

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

   4. fma-def 52.8%

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

   5. +-commutative 52.8%

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

   6. associate-+r- 52.7%

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

   7. fma-def 52.7%

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

3. Simplified 52.7%

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

4. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. metadata-eval 98.3%

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

   3. cancel-sign-sub-inv 98.3%

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

   4. fma-def 98.4%

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

   5. cancel-sign-sub-inv 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

   9. fma-def 98.4%

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

   10. mul-1-neg 98.4%

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

   11. *-commutative 98.4%

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

   12. distribute-rgt-neg-in 98.4%

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

   13. mul-1-neg 98.4%

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

   14. unsub-neg 98.4%

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

   15. unpow2 98.4%

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

   16. distribute-rgt-neg-in 98.4%

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

6. Simplified 98.4%

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

7. Taylor expanded in maxCos around 0 92.0%

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

   1. fma-def 92.0%

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

   2. unpow2 92.0%

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

9. Simplified 92.0%

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

10. Taylor expanded in uy around 0 74.4%

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

    1. *-commutative 74.4%

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

    2. associate-*l* 74.4%

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

    3. associate-*l* 74.4%

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

    4. +-commutative 74.4%

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

    5. *-commutative 74.4%

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

    6. mul-1-neg 74.4%

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

    7. unpow2 74.4%

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

    8. distribute-rgt-neg-out 74.4%

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

    9. distribute-lft-out 74.4%

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

    10. unsub-neg 74.4%

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

12. Simplified 74.4%

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

13. Taylor expanded in uy around 0 74.4%

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

    1. associate-*r* 74.4%

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

    2. *-commutative 74.4%

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

15. Simplified 74.4%

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

16. Final simplification 74.4%

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

Alternative 8: 63.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.8%

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

   1. associate-*l* 52.8%

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

   2. +-commutative 52.8%

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

   3. associate-+r- 52.8%

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

   4. fma-def 52.8%

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

   5. +-commutative 52.8%

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

   6. associate-+r- 52.7%

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

   7. fma-def 52.7%

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

3. Simplified 52.7%

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

4. Taylor expanded in ux around -inf 98.3%

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

   1. +-commutative 98.3%

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

   2. metadata-eval 98.3%

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

   3. cancel-sign-sub-inv 98.3%

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

   4. fma-def 98.4%

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

   5. cancel-sign-sub-inv 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. *-commutative 98.4%

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

   9. fma-def 98.4%

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

   10. mul-1-neg 98.4%

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

   11. *-commutative 98.4%

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

   12. distribute-rgt-neg-in 98.4%

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

   13. mul-1-neg 98.4%

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

   14. unsub-neg 98.4%

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

   15. unpow2 98.4%

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

   16. distribute-rgt-neg-in 98.4%

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

6. Simplified 98.4%

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

7. Taylor expanded in maxCos around 0 92.0%

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

   1. fma-def 92.0%

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

   2. unpow2 92.0%

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

9. Simplified 92.0%

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

10. Taylor expanded in uy around 0 74.4%

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

    1. *-commutative 74.4%

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

    2. associate-*l* 74.4%

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

    3. associate-*l* 74.4%

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

    4. +-commutative 74.4%

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

    5. *-commutative 74.4%

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

    6. mul-1-neg 74.4%

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

    7. unpow2 74.4%

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

    8. distribute-rgt-neg-out 74.4%

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

    9. distribute-lft-out 74.4%

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

    10. unsub-neg 74.4%

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

12. Simplified 74.4%

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

13. Taylor expanded in ux around 0 63.4%

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

14. Final simplification 63.4%

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

Reproduce

herbie shell --seed 2023279 
(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)))))))