UniformSampleCone, y

Percentage Accurate: 57.3% → 98.3%

Time: 30.8s

Alternatives: 10

Speedup: 1.5×

Specification

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-def 59.2%

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

   6. +-commutative 59.2%

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

   7. associate-+r- 59.1%

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

   8. fma-def 59.1%

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

   9. neg-sub0 59.1%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 59.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 59.0%

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

   14. +-commutative 59.0%

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

   15. sub-neg 59.0%

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

   16. fma-def 59.0%

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

3. Simplified 59.0%

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

4. Taylor expanded in ux around 0 98.2%

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

   1. +-commutative 98.2%

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

   2. fma-def 98.2%

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

   3. +-commutative 98.2%

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

   4. mul-1-neg 98.2%

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

   5. sub-neg 98.2%

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

   6. metadata-eval 98.2%

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

   7. distribute-neg-in 98.2%

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

   8. metadata-eval 98.2%

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

   9. +-commutative 98.2%

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

   10. mul-1-neg 98.2%

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

   11. associate--l+ 98.2%

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

   12. mul-1-neg 98.2%

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

   13. sub-neg 98.2%

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

   14. sub-neg 98.2%

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

   15. metadata-eval 98.2%

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

   16. *-commutative 98.2%

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

   17. unpow2 98.2%

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

6. Simplified 98.2%

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

   1. pow1 98.2%

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

   2. count-2 98.2%

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

   3. *-commutative 98.2%

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

8. Applied egg-rr 98.2%

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

   1. unpow1 98.2%

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

10. Simplified 98.2%

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

11. Final simplification 98.2%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-def 59.2%

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

   6. +-commutative 59.2%

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

   7. associate-+r- 59.1%

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

   8. fma-def 59.1%

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

   9. neg-sub0 59.1%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 59.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 59.0%

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

   14. +-commutative 59.0%

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

   15. sub-neg 59.0%

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

   16. fma-def 59.0%

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

3. Simplified 59.0%

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

4. Taylor expanded in ux around 0 98.2%

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

   1. +-commutative 98.2%

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

   2. fma-def 98.2%

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

   3. +-commutative 98.2%

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

   4. mul-1-neg 98.2%

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

   5. sub-neg 98.2%

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

   6. metadata-eval 98.2%

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

   7. distribute-neg-in 98.2%

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

   8. metadata-eval 98.2%

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

   9. +-commutative 98.2%

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

   10. mul-1-neg 98.2%

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

   11. associate--l+ 98.2%

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

   12. mul-1-neg 98.2%

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

   13. sub-neg 98.2%

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

   14. sub-neg 98.2%

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

   15. metadata-eval 98.2%

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

   16. *-commutative 98.2%

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

   17. unpow2 98.2%

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

6. Simplified 98.2%

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

   1. pow1 98.2%

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

   2. count-2 98.2%

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

   3. *-commutative 98.2%

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

8. Applied egg-rr 98.2%

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

   1. unpow1 98.2%

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

10. Simplified 98.2%

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

11. Taylor expanded in uy around inf 98.2%

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

    1. *-commutative 98.2%

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

    2. associate-*r* 98.2%

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

    3. *-commutative 98.2%

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

    4. *-commutative 98.2%

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

    5. *-commutative 98.2%

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

    6. *-commutative 98.2%

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

    7. *-commutative 98.2%

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

    8. unpow2 98.2%

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

    9. associate-*r* 98.2%

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

    10. associate-*r* 98.2%

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

    11. distribute-rgt-out 98.2%

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

    12. *-commutative 98.2%

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

    13. *-commutative 98.2%

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

    14. sub-neg 98.2%

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

    15. metadata-eval 98.2%

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

13. Simplified 98.2%

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

14. Final simplification 98.2%

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

Alternative 3: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.007600000128149986:\\ \;\;\;\;\left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.007600000128149986)
   (*
    (* PI (* uy 2.0))
    (sqrt (* (* ux (- 1.0 maxCos)) (+ 2.0 (* ux (+ maxCos -1.0))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.007600000128149986f) {
		tmp = (((float) M_PI) * (uy * 2.0f)) * sqrtf(((ux * (1.0f - maxCos)) * (2.0f + (ux * (maxCos + -1.0f)))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.007600000128149986))
		tmp = Float32(Float32(Float32(pi) * Float32(uy * Float32(2.0))) * sqrt(Float32(Float32(ux * Float32(Float32(1.0) - maxCos)) * Float32(Float32(2.0) + Float32(ux * Float32(maxCos + Float32(-1.0)))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.007600000128149986))
		tmp = (single(pi) * (uy * single(2.0))) * sqrt(((ux * (single(1.0) - maxCos)) * (single(2.0) + (ux * (maxCos + single(-1.0))))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00760000013

   1. Initial program 60.5%

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

      1. associate-*l* 60.5%

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

      2. sub-neg 60.5%

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

      3. +-commutative 60.5%

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

      4. distribute-rgt-neg-in 60.5%

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

      5. fma-def 60.5%

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

      6. +-commutative 60.5%

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

      7. associate-+r- 60.5%

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

      8. fma-def 60.5%

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

      9. neg-sub0 60.5%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

      10. +-commutative 60.5%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

      11. associate-+r- 60.4%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

      12. associate--r- 60.4%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

      13. neg-sub0 60.4%

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

      14. +-commutative 60.4%

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

      15. sub-neg 60.4%

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

      16. fma-def 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in ux around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. fma-def 98.5%

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

      3. +-commutative 98.5%

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

      4. mul-1-neg 98.5%

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

      5. sub-neg 98.5%

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

      6. metadata-eval 98.5%

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

      7. distribute-neg-in 98.5%

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

      8. metadata-eval 98.5%

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

      9. +-commutative 98.5%

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

      10. mul-1-neg 98.5%

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

      11. associate--l+ 98.5%

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

      12. mul-1-neg 98.5%

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

      13. sub-neg 98.5%

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

      14. sub-neg 98.5%

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

      15. metadata-eval 98.5%

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

      16. *-commutative 98.5%

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

      17. unpow2 98.5%

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

   6. Simplified 98.5%

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

      1. pow1 98.5%

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

      2. count-2 98.5%

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

      3. *-commutative 98.5%

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

   8. Applied egg-rr 98.5%

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

      1. unpow1 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in uy around 0 94.9%

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

       1. associate-*r* 94.9%

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

       2. associate-*r* 94.9%

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

       3. *-commutative 94.9%

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

       4. *-commutative 94.9%

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

       5. *-commutative 94.9%

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

   13. Simplified 94.9%

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

   if 0.00760000013 < (*.f32 uy 2)

   1. Initial program 55.1%

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

      1. associate-*l* 55.1%

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

      2. sub-neg 55.1%

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

      3. +-commutative 55.1%

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

      4. distribute-rgt-neg-in 55.1%

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

      5. fma-def 54.8%

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

      6. +-commutative 54.8%

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

      7. associate-+r- 54.7%

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

      8. fma-def 54.7%

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

      9. neg-sub0 54.7%

         \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

      10. +-commutative 54.7%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

      11. associate-+r- 54.7%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

      12. associate--r- 54.7%

          \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

      13. neg-sub0 54.7%

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

      14. +-commutative 54.7%

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

      15. sub-neg 54.7%

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

      16. fma-def 54.7%

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

   3. Simplified 54.7%

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

   4. Taylor expanded in maxCos around 0 51.7%

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

      1. +-commutative 51.7%

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

      2. fma-def 51.4%

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

      3. sub-neg 51.4%

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

      4. metadata-eval 51.4%

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

   6. Simplified 51.4%

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

   7. Taylor expanded in ux around 0 75.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.007600000128149986:\\ \;\;\;\;\left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(2 + ux \cdot \left(maxCos + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 4: 92.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-def 59.2%

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

   6. +-commutative 59.2%

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

   7. associate-+r- 59.1%

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

   8. fma-def 59.1%

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

   9. neg-sub0 59.1%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 59.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 59.0%

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

   14. +-commutative 59.0%

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

   15. sub-neg 59.0%

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

   16. fma-def 59.0%

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

3. Simplified 59.0%

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

4. Taylor expanded in ux around 0 98.2%

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

   1. +-commutative 98.2%

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

   2. fma-def 98.2%

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

   3. +-commutative 98.2%

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

   4. mul-1-neg 98.2%

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

   5. sub-neg 98.2%

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

   6. metadata-eval 98.2%

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

   7. distribute-neg-in 98.2%

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

   8. metadata-eval 98.2%

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

   9. +-commutative 98.2%

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

   10. mul-1-neg 98.2%

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

   11. associate--l+ 98.2%

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

   12. mul-1-neg 98.2%

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

   13. sub-neg 98.2%

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

   14. sub-neg 98.2%

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

   15. metadata-eval 98.2%

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

   16. *-commutative 98.2%

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

   17. unpow2 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in maxCos around 0 93.4%

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

   1. associate-*r* 93.4%

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

   2. +-commutative 93.4%

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

   3. mul-1-neg 93.4%

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

   4. unsub-neg 93.4%

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

   5. unpow2 93.4%

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

9. Simplified 93.4%

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

10. Final simplification 93.4%

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

Alternative 5: 92.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-def 59.2%

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

   6. +-commutative 59.2%

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

   7. associate-+r- 59.1%

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

   8. fma-def 59.1%

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

   9. neg-sub0 59.1%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 59.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 59.0%

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

   14. +-commutative 59.0%

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

   15. sub-neg 59.0%

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

   16. fma-def 59.0%

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

3. Simplified 59.0%

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

4. Taylor expanded in ux around 0 98.2%

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

   1. +-commutative 98.2%

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

   2. fma-def 98.2%

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

   3. +-commutative 98.2%

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

   4. mul-1-neg 98.2%

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

   5. sub-neg 98.2%

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

   6. metadata-eval 98.2%

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

   7. distribute-neg-in 98.2%

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

   8. metadata-eval 98.2%

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

   9. +-commutative 98.2%

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

   10. mul-1-neg 98.2%

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

   11. associate--l+ 98.2%

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

   12. mul-1-neg 98.2%

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

   13. sub-neg 98.2%

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

   14. sub-neg 98.2%

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

   15. metadata-eval 98.2%

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

   16. *-commutative 98.2%

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

   17. unpow2 98.2%

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

6. Simplified 98.2%

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

   1. pow1 98.2%

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

   2. count-2 98.2%

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

   3. *-commutative 98.2%

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

8. Applied egg-rr 98.2%

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

   1. unpow1 98.2%

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

10. Simplified 98.2%

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

11. Taylor expanded in uy around inf 98.2%

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

    1. *-commutative 98.2%

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

    2. associate-*r* 98.2%

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

    3. *-commutative 98.2%

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

    4. *-commutative 98.2%

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

    5. *-commutative 98.2%

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

    6. *-commutative 98.2%

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

    7. *-commutative 98.2%

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

    8. unpow2 98.2%

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

    9. associate-*r* 98.2%

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

    10. associate-*r* 98.2%

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

    11. distribute-rgt-out 98.2%

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

    12. *-commutative 98.2%

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

    13. *-commutative 98.2%

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

    14. sub-neg 98.2%

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

    15. metadata-eval 98.2%

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

13. Simplified 98.2%

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

14. Taylor expanded in maxCos around 0 93.3%

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

    1. *-commutative 93.3%

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

    2. associate-*r* 93.3%

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

    3. *-commutative 93.3%

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

    4. +-commutative 93.3%

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

    5. mul-1-neg 93.3%

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

    6. sub-neg 93.3%

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

    7. *-commutative 93.3%

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

16. Simplified 93.3%

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

17. Final simplification 93.3%

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

Alternative 6: 81.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-def 59.2%

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

   6. +-commutative 59.2%

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

   7. associate-+r- 59.1%

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

   8. fma-def 59.1%

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

   9. neg-sub0 59.1%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 59.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 59.0%

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

   14. +-commutative 59.0%

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

   15. sub-neg 59.0%

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

   16. fma-def 59.0%

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

3. Simplified 59.0%

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

4. Taylor expanded in ux around 0 98.2%

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

   1. +-commutative 98.2%

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

   2. fma-def 98.2%

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

   3. +-commutative 98.2%

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

   4. mul-1-neg 98.2%

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

   5. sub-neg 98.2%

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

   6. metadata-eval 98.2%

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

   7. distribute-neg-in 98.2%

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

   8. metadata-eval 98.2%

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

   9. +-commutative 98.2%

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

   10. mul-1-neg 98.2%

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

   11. associate--l+ 98.2%

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

   12. mul-1-neg 98.2%

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

   13. sub-neg 98.2%

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

   14. sub-neg 98.2%

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

   15. metadata-eval 98.2%

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

   16. *-commutative 98.2%

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

   17. unpow2 98.2%

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

6. Simplified 98.2%

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

   1. pow1 98.2%

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

   2. count-2 98.2%

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

   3. *-commutative 98.2%

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

8. Applied egg-rr 98.2%

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

   1. unpow1 98.2%

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

10. Simplified 98.2%

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

11. Taylor expanded in uy around 0 81.7%

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

    1. associate-*r* 81.7%

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

    2. associate-*r* 81.7%

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

    3. *-commutative 81.7%

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

    4. *-commutative 81.7%

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

    5. *-commutative 81.7%

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

13. Simplified 81.7%

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

14. Final simplification 81.7%

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

Alternative 7: 77.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-def 59.2%

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

   6. +-commutative 59.2%

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

   7. associate-+r- 59.1%

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

   8. fma-def 59.1%

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

   9. neg-sub0 59.1%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 59.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 59.0%

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

   14. +-commutative 59.0%

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

   15. sub-neg 59.0%

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

   16. fma-def 59.0%

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

3. Simplified 59.0%

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

4. Taylor expanded in uy around 0 51.8%

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

5. Taylor expanded in ux around 0 81.7%

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

   1. +-commutative 81.7%

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

   2. associate-*r* 81.7%

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

   3. *-commutative 81.7%

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

   4. associate-*r* 81.7%

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

   5. fma-def 81.7%

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

   6. +-commutative 81.7%

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

   7. mul-1-neg 81.7%

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

   8. sub-neg 81.7%

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

   9. metadata-eval 81.7%

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

   10. distribute-neg-in 81.7%

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

   11. metadata-eval 81.7%

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

   12. +-commutative 81.7%

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

   13. sub-neg 81.7%

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

   14. associate--l+ 81.7%

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

   15. unpow2 81.7%

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

   16. sub-neg 81.7%

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

   17. metadata-eval 81.7%

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

   18. associate-*r* 81.7%

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

7. Simplified 81.7%

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

8. Taylor expanded in maxCos around 0 78.5%

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

   1. +-commutative 78.5%

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

   2. mul-1-neg 78.5%

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

   3. unsub-neg 78.5%

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

   4. unpow2 78.5%

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

10. Simplified 78.5%

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

11. Final simplification 78.5%

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

Alternative 8: 77.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-def 59.2%

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

   6. +-commutative 59.2%

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

   7. associate-+r- 59.1%

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

   8. fma-def 59.1%

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

   9. neg-sub0 59.1%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 59.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 59.0%

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

   14. +-commutative 59.0%

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

   15. sub-neg 59.0%

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

   16. fma-def 59.0%

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

3. Simplified 59.0%

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

4. Taylor expanded in uy around 0 51.8%

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

5. Taylor expanded in ux around 0 81.7%

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

   1. +-commutative 81.7%

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

   2. associate-*r* 81.7%

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

   3. *-commutative 81.7%

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

   4. associate-*r* 81.7%

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

   5. fma-def 81.7%

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

   6. +-commutative 81.7%

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

   7. mul-1-neg 81.7%

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

   8. sub-neg 81.7%

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

   9. metadata-eval 81.7%

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

   10. distribute-neg-in 81.7%

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

   11. metadata-eval 81.7%

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

   12. +-commutative 81.7%

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

   13. sub-neg 81.7%

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

   14. associate--l+ 81.7%

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

   15. unpow2 81.7%

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

   16. sub-neg 81.7%

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

   17. metadata-eval 81.7%

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

   18. associate-*r* 81.7%

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

7. Simplified 81.7%

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

8. Taylor expanded in maxCos around 0 78.5%

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

   1. +-commutative 78.5%

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

   2. mul-1-neg 78.5%

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

   3. unsub-neg 78.5%

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

   4. unpow2 78.5%

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

10. Simplified 78.5%

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

11. Taylor expanded in uy around 0 78.5%

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

    1. *-commutative 78.5%

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

    2. *-commutative 78.5%

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

    3. unpow2 78.5%

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

    4. distribute-rgt-out-- 78.4%

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

    5. associate-*l* 78.3%

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

13. Simplified 78.3%

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

14. Final simplification 78.3%

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

Alternative 9: 77.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-def 59.2%

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

   6. +-commutative 59.2%

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

   7. associate-+r- 59.1%

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

   8. fma-def 59.1%

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

   9. neg-sub0 59.1%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 59.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 59.0%

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

   14. +-commutative 59.0%

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

   15. sub-neg 59.0%

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

   16. fma-def 59.0%

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

3. Simplified 59.0%

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

4. Taylor expanded in uy around 0 51.8%

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

5. Taylor expanded in ux around 0 81.7%

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

   1. +-commutative 81.7%

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

   2. associate-*r* 81.7%

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

   3. *-commutative 81.7%

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

   4. associate-*r* 81.7%

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

   5. fma-def 81.7%

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

   6. +-commutative 81.7%

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

   7. mul-1-neg 81.7%

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

   8. sub-neg 81.7%

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

   9. metadata-eval 81.7%

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

   10. distribute-neg-in 81.7%

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

   11. metadata-eval 81.7%

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

   12. +-commutative 81.7%

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

   13. sub-neg 81.7%

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

   14. associate--l+ 81.7%

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

   15. unpow2 81.7%

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

   16. sub-neg 81.7%

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

   17. metadata-eval 81.7%

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

   18. associate-*r* 81.7%

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

7. Simplified 81.7%

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

   1. expm1-log1p-u 81.7%

      \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), \left(maxCos + -1\right) \cdot \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot ux\right)\right)}\right)\right)} \cdot \left(uy \cdot \pi\right)\right) \]

   2. expm1-udef 71.1%

      \[\leadsto 2 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), \left(maxCos + -1\right) \cdot \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot ux\right)\right)}\right)} - 1\right)} \cdot \left(uy \cdot \pi\right)\right) \]

   3. count-2 71.1%

      \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(ux, \color{blue}{2 \cdot \left(1 - maxCos\right)}, \left(maxCos + -1\right) \cdot \left(\left(\left(1 - maxCos\right) \cdot ux\right) \cdot ux\right)\right)}\right)} - 1\right) \cdot \left(uy \cdot \pi\right)\right) \]

   4. associate-*r* 71.1%

      \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), \left(maxCos + -1\right) \cdot \color{blue}{\left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)}\right)}\right)} - 1\right) \cdot \left(uy \cdot \pi\right)\right) \]

   5. *-commutative 71.1%

      \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), \left(maxCos + -1\right) \cdot \color{blue}{\left(\left(ux \cdot ux\right) \cdot \left(1 - maxCos\right)\right)}\right)}\right)} - 1\right) \cdot \left(uy \cdot \pi\right)\right) \]

   6. associate-*l* 71.1%

      \[\leadsto 2 \cdot \left(\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), \left(maxCos + -1\right) \cdot \color{blue}{\left(ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)}\right)}\right)} - 1\right) \cdot \left(uy \cdot \pi\right)\right) \]

9. Applied egg-rr 71.1%

   \[\leadsto 2 \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), \left(maxCos + -1\right) \cdot \left(ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)}\right)} - 1\right)} \cdot \left(uy \cdot \pi\right)\right) \]
10. Step-by-step derivation

    1. expm1-def 81.7%

       \[\leadsto 2 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), \left(maxCos + -1\right) \cdot \left(ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)}\right)\right)} \cdot \left(uy \cdot \pi\right)\right) \]

    2. expm1-log1p 81.7%

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

    3. fma-udef 81.7%

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

    4. sub-neg 81.7%

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

    5. distribute-lft-in 81.7%

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

    6. metadata-eval 81.7%

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

    7. distribute-rgt-neg-in 81.7%

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

    8. sub-neg 81.7%

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

    9. *-commutative 81.7%

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

    10. metadata-eval 81.7%

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

    11. sub-neg 81.7%

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

    12. associate-*r* 81.7%

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

    13. unpow2 81.7%

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

    14. *-commutative 81.7%

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

11. Simplified 81.7%

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

12. Taylor expanded in maxCos around 0 78.5%

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

    1. +-commutative 78.5%

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

    2. mul-1-neg 78.5%

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

    3. sub-neg 78.5%

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

    4. unpow2 78.5%

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

    5. distribute-rgt-out-- 78.4%

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

14. Simplified 78.4%

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

15. Final simplification 78.4%

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

Alternative 10: 7.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{0}\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (* 2.0 (* (* uy PI) (sqrt 0.0))))

C

float code(float ux, float uy, float maxCos) {
	return 2.0f * ((uy * ((float) M_PI)) * sqrtf(0.0f));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(0.0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(2.0) * ((uy * single(pi)) * sqrt(single(0.0)));
end

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-def 59.2%

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

   6. +-commutative 59.2%

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

   7. associate-+r- 59.1%

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

   8. fma-def 59.1%

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

   9. neg-sub0 59.1%

      \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]

   10. +-commutative 59.1%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]

   11. associate-+r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]

   12. associate--r- 59.0%

       \[\leadsto \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]

   13. neg-sub0 59.0%

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

   14. +-commutative 59.0%

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

   15. sub-neg 59.0%

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

   16. fma-def 59.0%

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

3. Simplified 59.0%

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

4. Taylor expanded in uy around 0 51.8%

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

5. Taylor expanded in ux around 0 7.1%

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

6. Final simplification 7.1%

   \[\leadsto 2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{0}\right) \]

Reproduce

herbie shell --seed 2023187 
(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)))))))