UniformSampleCone, y

Percentage Accurate: 57.7% → 98.3%

Time: 19.9s

Alternatives: 8

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: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. associate-*l* 59.9%

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

   2. sub-neg 59.9%

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

   3. +-commutative 59.9%

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

   4. distribute-rgt-neg-in 59.9%

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

   5. fma-define 59.9%

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

3. Simplified 59.9%

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

5. Taylor expanded in ux around inf 98.0%

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

   1. associate--l+ 98.0%

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

   2. associate-*r/ 98.0%

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

   3. mul-1-neg 98.0%

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

   4. sub-neg 98.0%

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

   5. metadata-eval 98.0%

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

   6. +-commutative 98.0%

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

   7. distribute-neg-in 98.0%

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

   8. metadata-eval 98.0%

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

   9. sub-neg 98.0%

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

   10. *-commutative 98.0%

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

   11. sub-neg 98.0%

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

   12. mul-1-neg 98.0%

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

   13. fma-define 98.0%

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

   14. sub-neg 98.0%

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

   15. metadata-eval 98.0%

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

   16. +-commutative 98.0%

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

   17. mul-1-neg 98.0%

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

   18. sub-neg 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in ux around 0 98.1%

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

9. Final simplification 98.1%

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

Alternative 2: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. associate-*l* 59.9%

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

   2. sub-neg 59.9%

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

   3. +-commutative 59.9%

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

   4. distribute-rgt-neg-in 59.9%

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

   5. fma-define 59.9%

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

3. Simplified 59.9%

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

5. Taylor expanded in ux around inf 98.0%

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

   1. associate--l+ 98.0%

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

   2. associate-*r/ 98.0%

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

   3. mul-1-neg 98.0%

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

   4. sub-neg 98.0%

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

   5. metadata-eval 98.0%

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

   6. +-commutative 98.0%

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

   7. distribute-neg-in 98.0%

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

   8. metadata-eval 98.0%

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

   9. sub-neg 98.0%

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

   10. *-commutative 98.0%

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

   11. sub-neg 98.0%

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

   12. mul-1-neg 98.0%

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

   13. fma-define 98.0%

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

   14. sub-neg 98.0%

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

   15. metadata-eval 98.0%

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

   16. +-commutative 98.0%

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

   17. mul-1-neg 98.0%

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

   18. sub-neg 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in uy around inf 98.0%

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

   1. associate-*l* 98.0%

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

   2. +-commutative 98.0%

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

   3. associate--l+ 98.0%

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

   4. *-commutative 98.0%

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

   5. sub-neg 98.0%

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

   6. metadata-eval 98.0%

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

   7. associate-*r/ 98.0%

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

   8. metadata-eval 98.0%

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

   9. associate-*r/ 98.0%

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

   10. div-sub 98.0%

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

   11. cancel-sign-sub-inv 98.0%

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

   12. metadata-eval 98.0%

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

10. Simplified 98.0%

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

11. Final simplification 98.0%

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

Alternative 3: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right) - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 9.999999974752427e-7)
   (* (sin (* uy (* 2.0 PI))) (sqrt (* ux (- 2.0 ux))))
   (*
    2.0
    (*
     (sqrt
      (*
       ux
       (- (+ 2.0 (* ux (* (- 1.0 maxCos) (+ maxCos -1.0)))) (* 2.0 maxCos))))
     (* uy PI)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 9.999999974752427e-7f) {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = 2.0f * (sqrtf((ux * ((2.0f + (ux * ((1.0f - maxCos) * (maxCos + -1.0f)))) - (2.0f * maxCos)))) * (uy * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(9.999999974752427e-7))
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = Float32(Float32(2.0) * Float32(sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(ux * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))) - Float32(Float32(2.0) * maxCos)))) * Float32(uy * Float32(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(9.999999974752427e-7))
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = single(2.0) * (sqrt((ux * ((single(2.0) + (ux * ((single(1.0) - maxCos) * (maxCos + single(-1.0))))) - (single(2.0) * maxCos)))) * (uy * single(pi)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999997e-7

   1. Initial program 59.3%

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

      1. associate-*l* 59.3%

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

      2. sub-neg 59.3%

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

      3. +-commutative 59.3%

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

      4. distribute-rgt-neg-in 59.3%

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

      5. fma-define 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in ux around inf 98.0%

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

      1. associate--l+ 98.0%

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

      2. associate-*r/ 98.0%

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

      3. mul-1-neg 98.0%

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

      4. sub-neg 98.0%

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

      5. metadata-eval 98.0%

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

      6. +-commutative 98.0%

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

      7. distribute-neg-in 98.0%

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

      8. metadata-eval 98.0%

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

      9. sub-neg 98.0%

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

      10. *-commutative 98.0%

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

      11. sub-neg 98.0%

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

      12. mul-1-neg 98.0%

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

      13. fma-define 98.0%

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

      14. sub-neg 98.0%

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

      15. metadata-eval 98.0%

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

      16. +-commutative 98.0%

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

      17. mul-1-neg 98.0%

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

      18. sub-neg 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in ux around 0 98.1%

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

   9. Taylor expanded in maxCos around 0 97.9%

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

       1. neg-mul-1 97.9%

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

       2. unsub-neg 97.9%

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

   11. Simplified 97.9%

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

   if 9.99999997e-7 < maxCos

   1. Initial program 64.1%

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

      1. associate-*l* 64.1%

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

      2. sub-neg 64.1%

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

      3. +-commutative 64.1%

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

      4. distribute-rgt-neg-in 64.1%

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

      5. fma-define 64.1%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in ux around inf 98.2%

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

      1. associate--l+ 98.3%

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

      2. associate-*r/ 98.3%

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

      3. mul-1-neg 98.3%

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

      4. sub-neg 98.3%

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

      5. metadata-eval 98.3%

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

      6. +-commutative 98.3%

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

      7. distribute-neg-in 98.3%

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

      8. metadata-eval 98.3%

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

      9. sub-neg 98.3%

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

      10. *-commutative 98.3%

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

      11. sub-neg 98.3%

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

      12. mul-1-neg 98.3%

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

      13. fma-define 98.3%

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

      14. sub-neg 98.3%

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

      15. metadata-eval 98.3%

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

      16. +-commutative 98.3%

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

      17. mul-1-neg 98.3%

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

      18. sub-neg 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in ux around 0 98.3%

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

   9. Taylor expanded in uy around 0 91.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right) - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. associate-*l* 59.9%

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

   2. sub-neg 59.9%

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

   3. +-commutative 59.9%

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

   4. distribute-rgt-neg-in 59.9%

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

   5. fma-define 59.9%

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

3. Simplified 59.9%

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

5. Taylor expanded in ux around inf 98.0%

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

   1. associate--l+ 98.0%

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

   2. associate-*r/ 98.0%

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

   3. mul-1-neg 98.0%

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

   4. sub-neg 98.0%

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

   5. metadata-eval 98.0%

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

   6. +-commutative 98.0%

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

   7. distribute-neg-in 98.0%

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

   8. metadata-eval 98.0%

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

   9. sub-neg 98.0%

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

   10. *-commutative 98.0%

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

   11. sub-neg 98.0%

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

   12. mul-1-neg 98.0%

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

   13. fma-define 98.0%

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

   14. sub-neg 98.0%

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

   15. metadata-eval 98.0%

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

   16. +-commutative 98.0%

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

   17. mul-1-neg 98.0%

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

   18. sub-neg 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in uy around 0 81.8%

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

   1. associate-*l* 81.7%

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

   2. +-commutative 81.7%

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

   3. associate--l+ 81.8%

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

   4. *-commutative 81.8%

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

   5. sub-neg 81.8%

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

   6. metadata-eval 81.8%

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

   7. associate-*r/ 81.8%

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

   8. metadata-eval 81.8%

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

   9. associate-*r/ 81.8%

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

   10. div-sub 81.8%

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

   11. cancel-sign-sub-inv 81.8%

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

   12. metadata-eval 81.8%

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

10. Simplified 81.8%

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

11. Final simplification 81.8%

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

Alternative 5: 81.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. associate-*l* 59.9%

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

   2. sub-neg 59.9%

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

   3. +-commutative 59.9%

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

   4. distribute-rgt-neg-in 59.9%

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

   5. fma-define 59.9%

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

3. Simplified 59.9%

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

5. Taylor expanded in ux around inf 98.0%

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

   1. associate--l+ 98.0%

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

   2. associate-*r/ 98.0%

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

   3. mul-1-neg 98.0%

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

   4. sub-neg 98.0%

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

   5. metadata-eval 98.0%

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

   6. +-commutative 98.0%

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

   7. distribute-neg-in 98.0%

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

   8. metadata-eval 98.0%

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

   9. sub-neg 98.0%

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

   10. *-commutative 98.0%

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

   11. sub-neg 98.0%

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

   12. mul-1-neg 98.0%

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

   13. fma-define 98.0%

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

   14. sub-neg 98.0%

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

   15. metadata-eval 98.0%

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

   16. +-commutative 98.0%

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

   17. mul-1-neg 98.0%

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

   18. sub-neg 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in ux around 0 98.1%

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

9. Taylor expanded in uy around 0 81.9%

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

10. Final simplification 81.9%

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

Alternative 6: 65.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. associate-*l* 59.9%

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

   2. sub-neg 59.9%

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

   3. +-commutative 59.9%

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

   4. distribute-rgt-neg-in 59.9%

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

   5. fma-define 59.9%

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

3. Simplified 59.9%

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

5. Taylor expanded in uy around 0 52.5%

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

7. Simplified 52.6%

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

8. Taylor expanded in ux around 0 65.2%

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

9. Final simplification 65.2%

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

Alternative 7: 76.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. associate-*l* 59.9%

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

   2. sub-neg 59.9%

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

   3. +-commutative 59.9%

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

   4. distribute-rgt-neg-in 59.9%

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

   5. fma-define 59.9%

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

3. Simplified 59.9%

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

5. Taylor expanded in ux around inf 98.0%

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

   1. associate--l+ 98.0%

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

   2. associate-*r/ 98.0%

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

   3. mul-1-neg 98.0%

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

   4. sub-neg 98.0%

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

   5. metadata-eval 98.0%

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

   6. +-commutative 98.0%

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

   7. distribute-neg-in 98.0%

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

   8. metadata-eval 98.0%

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

   9. sub-neg 98.0%

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

   10. *-commutative 98.0%

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

   11. sub-neg 98.0%

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

   12. mul-1-neg 98.0%

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

   13. fma-define 98.0%

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

   14. sub-neg 98.0%

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

   15. metadata-eval 98.0%

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

   16. +-commutative 98.0%

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

   17. mul-1-neg 98.0%

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

   18. sub-neg 98.0%

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

7. Simplified 98.0%

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

8. Taylor expanded in maxCos around 0 93.3%

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

   1. pow1 93.3%

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

   2. associate-*l* 93.3%

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

   3. associate-*r* 93.3%

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

   4. *-commutative 93.3%

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

   5. associate-*r* 93.3%

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

   6. sub-neg 93.3%

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

   7. un-div-inv 93.3%

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

   8. metadata-eval 93.3%

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

10. Applied egg-rr 93.3%

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

    1. unpow1 93.3%

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

    2. *-commutative 93.3%

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

    3. metadata-eval 93.3%

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

    4. associate-*r/ 93.3%

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

    5. +-commutative 93.3%

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

    6. associate-*r/ 93.3%

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

    7. metadata-eval 93.3%

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

12. Simplified 93.3%

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

13. Taylor expanded in uy around 0 77.7%

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

    1. associate-*r* 77.7%

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

    2. sub-neg 77.7%

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

    3. metadata-eval 77.7%

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

    4. +-commutative 77.7%

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

    5. associate-*r/ 77.7%

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

    6. metadata-eval 77.7%

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

15. Simplified 77.7%

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

16. Final simplification 77.7%

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

Alternative 8: 62.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.9%

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

   1. associate-*l* 59.9%

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

   2. sub-neg 59.9%

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

   3. +-commutative 59.9%

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

   4. distribute-rgt-neg-in 59.9%

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

   5. fma-define 59.9%

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

3. Simplified 59.9%

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

5. Taylor expanded in uy around 0 52.5%

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

7. Simplified 52.6%

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

8. Taylor expanded in ux around 0 65.2%

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

9. Taylor expanded in maxCos around 0 62.9%

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

10. Final simplification 62.9%

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

Reproduce

herbie shell --seed 2024112 
(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)))))))