UniformSampleCone, y

Percentage Accurate: 57.4% → 98.4%

Time: 25.4s

Alternatives: 19

Speedup: 2.0×

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: 57.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.9%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. +-commutative 54.9%

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

   4. distribute-rgt-neg-in 54.9%

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

   5. fma-define 55.0%

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

3. Simplified 55.1%

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

5. Taylor expanded in ux around inf 98.4%

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

   1. fma-define 98.4%

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

   2. +-commutative 98.4%

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

   3. sub-neg 98.4%

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

   4. metadata-eval 98.4%

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

   5. distribute-lft-in 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. associate--l+ 98.4%

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

   9. mul-1-neg 98.4%

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

   10. sub-neg 98.4%

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

   11. *-commutative 98.4%

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

   12. sub-neg 98.4%

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

   13. metadata-eval 98.4%

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

   14. +-commutative 98.4%

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

7. Simplified 98.4%

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

   1. *-commutative 98.4%

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

   2. add-cbrt-cube 98.4%

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

   3. add-cbrt-cube 98.4%

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

   4. cbrt-unprod 98.3%

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

9. Applied egg-rr 98.6%

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

10. Final simplification 98.6%

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

Alternative 2: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.9%

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

   1. add-cube-cbrt 54.8%

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

   2. pow3 54.8%

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

   3. pow2 54.8%

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

   4. +-commutative 54.8%

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

   5. fma-undefine 54.8%

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

4. Applied egg-rr 54.8%

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

5. Taylor expanded in ux around -inf 98.2%

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

   1. +-commutative 98.2%

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

   2. mul-1-neg 98.2%

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

   3. unsub-neg 98.2%

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

   4. metadata-eval 98.2%

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

   5. cancel-sign-sub-inv 98.2%

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

   6. cancel-sign-sub-inv 98.2%

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

   7. metadata-eval 98.2%

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

   8. *-commutative 98.2%

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

   9. mul-1-neg 98.2%

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

   10. sub-neg 98.2%

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

7. Simplified 98.2%

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

   1. *-commutative 98.2%

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

   2. add-cbrt-cube 98.2%

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

   3. *-commutative 98.2%

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

   4. rem-cbrt-cube 98.2%

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

   5. cbrt-unprod 98.1%

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

9. Applied egg-rr 98.5%

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

10. Final simplification 98.5%

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

Alternative 3: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.9%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. +-commutative 54.9%

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

   4. distribute-rgt-neg-in 54.9%

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

   5. fma-define 55.0%

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

3. Simplified 55.1%

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

5. Taylor expanded in ux around inf 98.4%

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

   1. fma-define 98.4%

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

   2. +-commutative 98.4%

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

   3. sub-neg 98.4%

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

   4. metadata-eval 98.4%

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

   5. distribute-lft-in 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. associate--l+ 98.4%

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

   9. mul-1-neg 98.4%

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

   10. sub-neg 98.4%

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

   11. *-commutative 98.4%

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

   12. sub-neg 98.4%

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

   13. metadata-eval 98.4%

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

   14. +-commutative 98.4%

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

7. Simplified 98.4%

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

8. Taylor expanded in uy around inf 98.4%

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

10. Simplified 98.4%

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

11. Final simplification 98.4%

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

Alternative 4: 95.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + maxCos \cdot -2, {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)} \cdot \left(2 \cdot \left(\pi \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.999999873689376e-5)
   (* (sin (* uy (* 2.0 PI))) (sqrt (- (* ux 2.0) (pow ux 2.0))))
   (*
    (sqrt
     (fma
      ux
      (+ 2.0 (* maxCos -2.0))
      (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))))
    (* 2.0 (* PI uy)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.999999873689376e-5f) {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((ux * 2.0f) - powf(ux, 2.0f)));
	} else {
		tmp = sqrtf(fmaf(ux, (2.0f + (maxCos * -2.0f)), (powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f))))) * (2.0f * (((float) M_PI) * uy));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.999999873689376e-5))
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(ux * Float32(2.0)) - (ux ^ Float32(2.0)))));
	else
		tmp = Float32(sqrt(fma(ux, Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))), Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0)))))) * Float32(Float32(2.0) * Float32(Float32(pi) * uy)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999987e-5

   1. Initial program 54.8%

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

      1. associate-*l* 54.8%

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

      2. sub-neg 54.8%

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

      3. +-commutative 54.8%

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

      4. distribute-rgt-neg-in 54.8%

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

      5. fma-define 55.0%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in ux around inf 98.4%

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

      1. fma-define 98.4%

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

      2. +-commutative 98.4%

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

      3. sub-neg 98.4%

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

      4. metadata-eval 98.4%

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

      5. distribute-lft-in 98.4%

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

      6. metadata-eval 98.4%

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

      7. +-commutative 98.4%

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

      8. associate--l+ 98.4%

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

      9. mul-1-neg 98.4%

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

      10. sub-neg 98.4%

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

      11. *-commutative 98.4%

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

      12. sub-neg 98.4%

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

      13. metadata-eval 98.4%

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

      14. +-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in maxCos around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. mul-1-neg 97.6%

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

      3. unsub-neg 97.6%

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

   10. Simplified 97.6%

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

   if 4.99999987e-5 < maxCos

   1. Initial program 55.4%

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

      1. associate-*l* 55.4%

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

      2. sub-neg 55.4%

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

      3. +-commutative 55.4%

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

      4. distribute-rgt-neg-in 55.4%

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

      5. fma-define 54.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in ux around inf 98.5%

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

      1. fma-define 98.6%

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

      2. +-commutative 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

      5. distribute-lft-in 98.6%

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

      6. metadata-eval 98.6%

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

      7. +-commutative 98.6%

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

      8. associate--l+ 98.9%

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

      9. mul-1-neg 98.9%

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

      10. sub-neg 98.9%

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

      11. *-commutative 98.9%

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

      12. sub-neg 98.9%

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

      13. metadata-eval 98.9%

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

      14. +-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in uy around 0 83.4%

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

   10. Simplified 83.5%

       \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 2 + maxCos \cdot -2, {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + maxCos \cdot -2, {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)} \cdot \left(2 \cdot \left(\pi \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.9%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. +-commutative 54.9%

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

   4. distribute-rgt-neg-in 54.9%

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

   5. fma-define 55.0%

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

3. Simplified 55.1%

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

5. Taylor expanded in ux around inf 98.4%

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

   1. fma-define 98.4%

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

   2. +-commutative 98.4%

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

   3. sub-neg 98.4%

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

   4. metadata-eval 98.4%

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

   5. distribute-lft-in 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. associate--l+ 98.4%

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

   9. mul-1-neg 98.4%

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

   10. sub-neg 98.4%

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

   11. *-commutative 98.4%

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

   12. sub-neg 98.4%

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

   13. metadata-eval 98.4%

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

   14. +-commutative 98.4%

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

7. Simplified 98.4%

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

8. Taylor expanded in uy around inf 98.4%

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

9. Final simplification 98.4%

   \[\leadsto \sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right) + {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)} \]
10. Add Preprocessing

Alternative 6: 92.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.9%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. +-commutative 54.9%

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

   4. distribute-rgt-neg-in 54.9%

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

   5. fma-define 55.0%

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

3. Simplified 55.1%

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

5. Taylor expanded in ux around inf 98.4%

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

   1. fma-define 98.4%

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

   2. +-commutative 98.4%

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

   3. sub-neg 98.4%

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

   4. metadata-eval 98.4%

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

   5. distribute-lft-in 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. associate--l+ 98.4%

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

   9. mul-1-neg 98.4%

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

   10. sub-neg 98.4%

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

   11. *-commutative 98.4%

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

   12. sub-neg 98.4%

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

   13. metadata-eval 98.4%

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

   14. +-commutative 98.4%

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

7. Simplified 98.4%

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

8. Taylor expanded in maxCos around 0 93.1%

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

   1. +-commutative 93.1%

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

   2. mul-1-neg 93.1%

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

   3. unsub-neg 93.1%

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

10. Simplified 93.1%

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

11. Final simplification 93.1%

    \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}} \]
12. Add Preprocessing

Alternative 7: 91.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ t_1 := 1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)\\ \mathbf{if}\;t\_1 \leq 0.00039999998989515007:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot 2 + -2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* PI (* 2.0 uy))))
        (t_1
         (+
          1.0
          (* (+ (- 1.0 ux) (* ux maxCos)) (- (+ ux -1.0) (* ux maxCos))))))
   (if (<= t_1 0.00039999998989515007)
     (* t_0 (sqrt (+ (* ux 2.0) (* -2.0 (* ux maxCos)))))
     (* t_0 (sqrt t_1)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf((((float) M_PI) * (2.0f * uy)));
	float t_1 = 1.0f + (((1.0f - ux) + (ux * maxCos)) * ((ux + -1.0f) - (ux * maxCos)));
	float tmp;
	if (t_1 <= 0.00039999998989515007f) {
		tmp = t_0 * sqrtf(((ux * 2.0f) + (-2.0f * (ux * maxCos))));
	} else {
		tmp = t_0 * sqrtf(t_1);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))
	t_1 = Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos)) * Float32(Float32(ux + Float32(-1.0)) - Float32(ux * maxCos))))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.00039999998989515007))
		tmp = Float32(t_0 * sqrt(Float32(Float32(ux * Float32(2.0)) + Float32(Float32(-2.0) * Float32(ux * maxCos)))));
	else
		tmp = Float32(t_0 * sqrt(t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = sin((single(pi) * (single(2.0) * uy)));
	t_1 = single(1.0) + (((single(1.0) - ux) + (ux * maxCos)) * ((ux + single(-1.0)) - (ux * maxCos)));
	tmp = single(0.0);
	if (t_1 <= single(0.00039999998989515007))
		tmp = t_0 * sqrt(((ux * single(2.0)) + (single(-2.0) * (ux * maxCos))));
	else
		tmp = t_0 * sqrt(t_1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 (*.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)))) < 3.9999999e-4

   1. Initial program 36.8%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0 40.3%

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

   4. Taylor expanded in maxCos around 0 92.7%

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

   if 3.9999999e-4 < (-.f32 1 (*.f32 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) (+.f32 (-.f32 1 ux) (*.f32 ux maxCos))))

   1. Initial program 88.2%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right) \leq 0.00039999998989515007:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2 + -2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot uy\right)\\ \mathbf{if}\;2 \cdot uy \leq 0.0012000000569969416:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{ux \cdot 2 + -2 \cdot \left(ux \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy))))
   (if (<= (* 2.0 uy) 0.0012000000569969416)
     (* t_0 (sqrt (- (* ux 2.0) (pow ux 2.0))))
     (* (sin t_0) (sqrt (+ (* ux 2.0) (* -2.0 (* ux maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (2.0f * uy);
	float tmp;
	if ((2.0f * uy) <= 0.0012000000569969416f) {
		tmp = t_0 * sqrtf(((ux * 2.0f) - powf(ux, 2.0f)));
	} else {
		tmp = sinf(t_0) * sqrtf(((ux * 2.0f) + (-2.0f * (ux * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0012000000569969416))
		tmp = Float32(t_0 * sqrt(Float32(Float32(ux * Float32(2.0)) - (ux ^ Float32(2.0)))));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(Float32(ux * Float32(2.0)) + Float32(Float32(-2.0) * Float32(ux * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(pi) * (single(2.0) * uy);
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.0012000000569969416))
		tmp = t_0 * sqrt(((ux * single(2.0)) - (ux ^ single(2.0))));
	else
		tmp = sin(t_0) * sqrt(((ux * single(2.0)) + (single(-2.0) * (ux * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00120000006

   1. Initial program 54.3%

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

      1. associate-*l* 54.3%

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

      2. sub-neg 54.3%

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

      3. +-commutative 54.3%

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

      4. distribute-rgt-neg-in 54.3%

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

      5. fma-define 54.4%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in ux around inf 98.5%

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

      1. fma-define 98.6%

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

      2. +-commutative 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

      5. distribute-lft-in 98.6%

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

      6. metadata-eval 98.6%

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

      7. +-commutative 98.6%

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

      8. associate--l+ 98.6%

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

      9. mul-1-neg 98.6%

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

      10. sub-neg 98.6%

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

      11. *-commutative 98.6%

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

      12. sub-neg 98.6%

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

      13. metadata-eval 98.6%

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

      14. +-commutative 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in maxCos around 0 93.5%

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

      1. +-commutative 93.5%

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

      2. mul-1-neg 93.5%

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

      3. unsub-neg 93.5%

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

   10. Simplified 93.5%

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

   11. Taylor expanded in uy around 0 92.7%

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

       1. associate-*r* 92.7%

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

       2. *-commutative 92.7%

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

       3. *-commutative 92.7%

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

       4. associate-*r* 92.7%

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

   13. Simplified 92.7%

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

   if 0.00120000006 < (*.f32 uy 2)

   1. Initial program 56.0%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0 46.9%

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

   4. Taylor expanded in maxCos around 0 78.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0012000000569969416:\\ \;\;\;\;\left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2 + -2 \cdot \left(ux \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot uy\right)\\ \mathbf{if}\;2 \cdot uy \leq 0.0012000000569969416:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy))))
   (if (<= (* 2.0 uy) 0.0012000000569969416)
     (* t_0 (sqrt (- (* ux 2.0) (pow ux 2.0))))
     (* (sin t_0) (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (2.0f * uy);
	float tmp;
	if ((2.0f * uy) <= 0.0012000000569969416f) {
		tmp = t_0 * sqrtf(((ux * 2.0f) - powf(ux, 2.0f)));
	} else {
		tmp = sinf(t_0) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0012000000569969416))
		tmp = Float32(t_0 * sqrt(Float32(Float32(ux * Float32(2.0)) - (ux ^ Float32(2.0)))));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(pi) * (single(2.0) * uy);
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.0012000000569969416))
		tmp = t_0 * sqrt(((ux * single(2.0)) - (ux ^ single(2.0))));
	else
		tmp = sin(t_0) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00120000006

   1. Initial program 54.3%

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

      1. associate-*l* 54.3%

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

      2. sub-neg 54.3%

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

      3. +-commutative 54.3%

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

      4. distribute-rgt-neg-in 54.3%

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

      5. fma-define 54.4%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in ux around inf 98.5%

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

      1. fma-define 98.6%

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

      2. +-commutative 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

      5. distribute-lft-in 98.6%

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

      6. metadata-eval 98.6%

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

      7. +-commutative 98.6%

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

      8. associate--l+ 98.6%

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

      9. mul-1-neg 98.6%

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

      10. sub-neg 98.6%

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

      11. *-commutative 98.6%

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

      12. sub-neg 98.6%

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

      13. metadata-eval 98.6%

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

      14. +-commutative 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in maxCos around 0 93.5%

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

      1. +-commutative 93.5%

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

      2. mul-1-neg 93.5%

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

      3. unsub-neg 93.5%

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

   10. Simplified 93.5%

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

   11. Taylor expanded in uy around 0 92.7%

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

       1. associate-*r* 92.7%

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

       2. *-commutative 92.7%

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

       3. *-commutative 92.7%

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

       4. associate-*r* 92.7%

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

   13. Simplified 92.7%

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

   if 0.00120000006 < (*.f32 uy 2)

   1. Initial program 56.0%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0 78.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0012000000569969416:\\ \;\;\;\;\left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00019999999494757503)
   (* (sin (* PI (* 2.0 uy))) (sqrt (+ (* ux 2.0) (* -2.0 (* ux maxCos)))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0)))))))

C

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

Julia

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

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = sin((single(pi) * (single(2.0) * uy))) * sqrt(((ux * single(2.0)) + (single(-2.0) * (ux * maxCos))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 36.6%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0 40.1%

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

   4. Taylor expanded in maxCos around 0 92.8%

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

   if 1.99999995e-4 < ux

   1. Initial program 87.9%

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

      1. associate-*l* 87.9%

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

      2. sub-neg 87.9%

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

      3. +-commutative 87.9%

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

      4. distribute-rgt-neg-in 87.9%

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

      5. fma-define 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in maxCos around 0 82.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2 + -2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot uy\right)\\ \mathbf{if}\;2 \cdot uy \leq 0.004000000189989805:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy))))
   (if (<= (* 2.0 uy) 0.004000000189989805)
     (* t_0 (sqrt (- (* ux 2.0) (pow ux 2.0))))
     (* (sin t_0) (sqrt (* ux 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (2.0f * uy);
	float tmp;
	if ((2.0f * uy) <= 0.004000000189989805f) {
		tmp = t_0 * sqrtf(((ux * 2.0f) - powf(ux, 2.0f)));
	} else {
		tmp = sinf(t_0) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.004000000189989805))
		tmp = Float32(t_0 * sqrt(Float32(Float32(ux * Float32(2.0)) - (ux ^ Float32(2.0)))));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(ux * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(pi) * (single(2.0) * uy);
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.004000000189989805))
		tmp = t_0 * sqrt(((ux * single(2.0)) - (ux ^ single(2.0))));
	else
		tmp = sin(t_0) * sqrt((ux * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00400000019

   1. Initial program 54.6%

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

      1. associate-*l* 54.6%

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

      2. sub-neg 54.6%

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

      3. +-commutative 54.6%

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

      4. distribute-rgt-neg-in 54.6%

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

      5. fma-define 54.7%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in ux around inf 98.5%

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

      1. fma-define 98.5%

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

      2. +-commutative 98.5%

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

      3. sub-neg 98.5%

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

      4. metadata-eval 98.5%

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

      5. distribute-lft-in 98.5%

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

      6. metadata-eval 98.5%

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

      7. +-commutative 98.5%

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

      8. associate--l+ 98.5%

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

      9. mul-1-neg 98.5%

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

      10. sub-neg 98.5%

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

      11. *-commutative 98.5%

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

      12. sub-neg 98.5%

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

      13. metadata-eval 98.5%

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

      14. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in maxCos around 0 93.2%

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

      1. +-commutative 93.2%

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

      2. mul-1-neg 93.2%

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

      3. unsub-neg 93.2%

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

   10. Simplified 93.2%

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

   11. Taylor expanded in uy around 0 91.0%

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

       1. associate-*r* 91.0%

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

       2. *-commutative 91.0%

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

       3. *-commutative 91.0%

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

       4. associate-*r* 91.0%

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

   13. Simplified 91.0%

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

   if 0.00400000019 < (*.f32 uy 2)

   1. Initial program 55.6%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0 47.4%

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

   4. Taylor expanded in maxCos around 0 75.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.004000000189989805:\\ \;\;\;\;\left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0004600000102072954:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0004600000102072954)
   (* (sin (* PI (* 2.0 uy))) (sqrt (* ux 2.0)))
   (*
    2.0
    (*
     (* PI uy)
     (sqrt
      (+
       1.0
       (* (+ 1.0 (- (* ux maxCos) ux)) (+ -1.0 (* ux (- 1.0 maxCos))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0004600000102072954f) {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * 2.0f));
	} else {
		tmp = 2.0f * ((((float) M_PI) * uy) * sqrtf((1.0f + ((1.0f + ((ux * maxCos) - ux)) * (-1.0f + (ux * (1.0f - maxCos)))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0004600000102072954))
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(2.0))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(Float32(pi) * uy) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(Float32(ux * maxCos) - ux)) * Float32(Float32(-1.0) + Float32(ux * Float32(Float32(1.0) - maxCos))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.0004600000102072954))
		tmp = sin((single(pi) * (single(2.0) * uy))) * sqrt((ux * single(2.0)));
	else
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((single(1.0) + ((single(1.0) + ((ux * maxCos) - ux)) * (single(-1.0) + (ux * (single(1.0) - maxCos)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 4.6000001e-4

   1. Initial program 39.1%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around 0 41.9%

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

   4. Taylor expanded in maxCos around 0 87.5%

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

   if 4.6000001e-4 < ux

   1. Initial program 89.6%

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

      1. associate-*l* 89.6%

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

      2. sub-neg 89.6%

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

      3. +-commutative 89.6%

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

      4. distribute-rgt-neg-in 89.6%

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

      5. fma-define 89.8%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in uy around 0 76.5%

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

   7. Simplified 76.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0004600000102072954:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos))));
	else
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((single(1.0) + ((single(1.0) - (ux * (single(1.0) - maxCos))) * (ux + (single(-1.0) - (ux * maxCos)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 36.6%

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

      1. associate-*l* 36.6%

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

      2. sub-neg 36.6%

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

      3. +-commutative 36.6%

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

      4. distribute-rgt-neg-in 36.6%

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

      5. fma-define 36.7%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in uy around 0 34.2%

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

   7. Simplified 34.2%

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

      1. +-commutative 34.2%

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

      2. distribute-rgt-neg-in 34.2%

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

      3. fma-define 34.2%

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

      4. +-commutative 34.2%

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

      5. fma-undefine 34.2%

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

      6. +-commutative 34.2%

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

   9. Applied egg-rr 34.2%

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

   10. Taylor expanded in ux around 0 79.1%

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

   if 1.99999995e-4 < ux

   1. Initial program 87.9%

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

      1. associate-*l* 87.9%

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

      2. sub-neg 87.9%

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

      3. +-commutative 87.9%

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

      4. distribute-rgt-neg-in 87.9%

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

      5. fma-define 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in uy around 0 74.2%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in uy around 0 74.2%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{1 + \left(1 - ux \cdot \left(1 - maxCos\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos))));
	else
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((single(1.0) + ((single(1.0) + ((ux * maxCos) - ux)) * (single(-1.0) + (ux * (single(1.0) - maxCos)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 36.6%

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

      1. associate-*l* 36.6%

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

      2. sub-neg 36.6%

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

      3. +-commutative 36.6%

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

      4. distribute-rgt-neg-in 36.6%

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

      5. fma-define 36.7%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in uy around 0 34.2%

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

   7. Simplified 34.2%

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

      1. +-commutative 34.2%

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

      2. distribute-rgt-neg-in 34.2%

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

      3. fma-define 34.2%

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

      4. +-commutative 34.2%

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

      5. fma-undefine 34.2%

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

      6. +-commutative 34.2%

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

   9. Applied egg-rr 34.2%

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

   10. Taylor expanded in ux around 0 79.1%

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

   if 1.99999995e-4 < ux

   1. Initial program 87.9%

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

      1. associate-*l* 87.9%

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

      2. sub-neg 87.9%

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

      3. +-commutative 87.9%

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

      4. distribute-rgt-neg-in 87.9%

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

      5. fma-define 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in uy around 0 74.2%

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

   7. Simplified 74.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos)))));
	else
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 36.6%

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

      1. associate-*l* 36.6%

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

      2. sub-neg 36.6%

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

      3. +-commutative 36.6%

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

      4. distribute-rgt-neg-in 36.6%

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

      5. fma-define 36.7%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in uy around 0 34.2%

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

   7. Simplified 34.2%

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

   8. Taylor expanded in ux around 0 79.1%

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

   if 1.99999995e-4 < ux

   1. Initial program 87.9%

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

      1. associate-*l* 87.9%

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

      2. sub-neg 87.9%

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

      3. +-commutative 87.9%

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

      4. distribute-rgt-neg-in 87.9%

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

      5. fma-define 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in uy around 0 74.2%

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

   7. Simplified 74.4%

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

      1. +-commutative 74.4%

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

      2. distribute-rgt-neg-in 74.4%

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

      3. fma-define 74.7%

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

      4. +-commutative 74.7%

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

      5. fma-undefine 74.7%

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

      6. +-commutative 74.7%

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

   9. Applied egg-rr 74.7%

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

   10. Taylor expanded in maxCos around 0 71.4%

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

       1. neg-mul-1 71.4%

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

       2. unsub-neg 71.4%

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

       3. sub-neg 71.4%

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

       4. metadata-eval 71.4%

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

   12. Simplified 71.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos)))));
	else
		tmp = single(2.0) * (uy * (single(pi) * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 36.6%

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

      1. associate-*l* 36.6%

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

      2. sub-neg 36.6%

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

      3. +-commutative 36.6%

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

      4. distribute-rgt-neg-in 36.6%

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

      5. fma-define 36.7%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in uy around 0 34.2%

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

   7. Simplified 34.2%

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

   8. Taylor expanded in ux around 0 79.1%

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

   if 1.99999995e-4 < ux

   1. Initial program 87.9%

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

      1. associate-*l* 87.9%

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

      2. sub-neg 87.9%

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

      3. +-commutative 87.9%

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

      4. distribute-rgt-neg-in 87.9%

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

      5. fma-define 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in uy around 0 74.2%

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

   7. Simplified 74.4%

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

      1. +-commutative 74.4%

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

      2. distribute-rgt-neg-in 74.4%

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

      3. fma-define 74.7%

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

      4. +-commutative 74.7%

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

      5. fma-undefine 74.7%

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

      6. +-commutative 74.7%

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

   9. Applied egg-rr 74.7%

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

   10. Taylor expanded in maxCos around 0 71.4%

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

       1. associate-*l* 71.5%

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

       2. neg-mul-1 71.5%

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

       3. sub-neg 71.5%

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

       4. metadata-eval 71.5%

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

   12. Simplified 71.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 75.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = single(2.0) * ((single(pi) * uy) * sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos))));
	else
		tmp = single(2.0) * (uy * (single(pi) * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 36.6%

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

      1. associate-*l* 36.6%

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

      2. sub-neg 36.6%

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

      3. +-commutative 36.6%

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

      4. distribute-rgt-neg-in 36.6%

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

      5. fma-define 36.7%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in uy around 0 34.2%

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

   7. Simplified 34.2%

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

      1. +-commutative 34.2%

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

      2. distribute-rgt-neg-in 34.2%

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

      3. fma-define 34.2%

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

      4. +-commutative 34.2%

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

      5. fma-undefine 34.2%

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

      6. +-commutative 34.2%

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

   9. Applied egg-rr 34.2%

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

   10. Taylor expanded in ux around 0 79.1%

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

   if 1.99999995e-4 < ux

   1. Initial program 87.9%

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

      1. associate-*l* 87.9%

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

      2. sub-neg 87.9%

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

      3. +-commutative 87.9%

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

      4. distribute-rgt-neg-in 87.9%

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

      5. fma-define 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in uy around 0 74.2%

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

   7. Simplified 74.4%

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

      1. +-commutative 74.4%

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

      2. distribute-rgt-neg-in 74.4%

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

      3. fma-define 74.7%

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

      4. +-commutative 74.7%

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

      5. fma-undefine 74.7%

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

      6. +-commutative 74.7%

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

   9. Applied egg-rr 74.7%

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

   10. Taylor expanded in maxCos around 0 71.4%

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

       1. associate-*l* 71.5%

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

       2. neg-mul-1 71.5%

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

       3. sub-neg 71.5%

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

       4. metadata-eval 71.5%

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

   12. Simplified 71.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.9%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. +-commutative 54.9%

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

   4. distribute-rgt-neg-in 54.9%

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

   5. fma-define 55.0%

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

3. Simplified 55.1%

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

5. Taylor expanded in uy around 0 48.4%

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

7. Simplified 48.5%

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

8. Taylor expanded in ux around 0 69.0%

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

9. Final simplification 69.0%

   \[\leadsto 2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \]
10. Add Preprocessing

Alternative 19: 63.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.9%

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

   1. associate-*l* 54.9%

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

   2. sub-neg 54.9%

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

   3. +-commutative 54.9%

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

   4. distribute-rgt-neg-in 54.9%

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

   5. fma-define 55.0%

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

3. Simplified 55.1%

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

5. Taylor expanded in uy around 0 48.4%

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

7. Simplified 48.5%

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

8. Taylor expanded in ux around 0 69.0%

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

9. Taylor expanded in maxCos around 0 66.4%

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

    1. *-commutative 66.4%

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

11. Simplified 66.4%

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

12. Final simplification 66.4%

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

Reproduce

herbie shell --seed 2024053 
(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)))))))