UniformSampleCone, x

Percentage Accurate: 56.6% → 99.0%

Time: 15.3s

Alternatives: 15

Speedup: 3.1×

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

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

FPCore

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

Initial Program: 56.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.3%

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

   1. associate-*l* 55.3%

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

   2. sub-neg 55.3%

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

   3. +-commutative 55.3%

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

   4. distribute-rgt-neg-in 55.3%

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

   5. fma-def 55.5%

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

3. Simplified 55.5%

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

4. Taylor expanded in ux around 0 98.9%

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

   1. fma-def 98.9%

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

   2. +-commutative 98.9%

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

   3. sub-neg 98.9%

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

   4. metadata-eval 98.9%

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

   5. +-commutative 98.9%

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

   6. distribute-lft-in 98.9%

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

   7. metadata-eval 98.9%

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

   8. associate--l+ 99.0%

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

   9. mul-1-neg 99.0%

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

   10. sub-neg 99.0%

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

   11. *-commutative 99.0%

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

   12. *-commutative 99.0%

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

   13. sub-neg 99.0%

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

   14. mul-1-neg 99.0%

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

   15. associate-*l* 99.0%

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

   16. sub-neg 99.0%

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

   17. metadata-eval 99.0%

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

   18. +-commutative 99.0%

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

   19. mul-1-neg 99.0%

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

   20. sub-neg 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in uy around inf 98.9%

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

   1. *-commutative 98.9%

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

   2. fma-def 99.0%

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

   3. cancel-sign-sub-inv 99.0%

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

   4. metadata-eval 99.0%

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

   5. *-commutative 99.0%

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

   6. sub-neg 99.0%

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

   7. mul-1-neg 99.0%

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

   8. mul-1-neg 99.0%

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

   9. sub-neg 99.0%

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

   10. sub-neg 99.0%

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

   11. metadata-eval 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.3%

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

   1. associate-*l* 55.3%

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

   2. sub-neg 55.3%

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

   3. +-commutative 55.3%

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

   4. distribute-rgt-neg-in 55.3%

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

   5. fma-def 55.5%

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

3. Simplified 55.5%

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

4. Taylor expanded in ux around -inf 98.9%

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

   1. +-commutative 98.9%

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

   2. mul-1-neg 98.9%

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

   3. unsub-neg 98.9%

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

   4. *-commutative 98.9%

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

   5. *-commutative 98.9%

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

   6. associate-*l* 98.9%

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

   7. sub-neg 98.9%

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

   8. metadata-eval 98.9%

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

   9. +-commutative 98.9%

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

   10. mul-1-neg 98.9%

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

   11. sub-neg 98.9%

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

   12. sub-neg 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in uy around inf 98.9%

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

8. Final simplification 98.9%

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

Alternative 3: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.3%

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

   1. associate-*l* 55.3%

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

   2. sub-neg 55.3%

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

   3. +-commutative 55.3%

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

   4. distribute-rgt-neg-in 55.3%

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

   5. fma-def 55.5%

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

3. Simplified 55.5%

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

4. Taylor expanded in ux around -inf 98.9%

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

   1. +-commutative 98.9%

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

   2. mul-1-neg 98.9%

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

   3. unsub-neg 98.9%

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

   4. *-commutative 98.9%

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

   5. *-commutative 98.9%

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

   6. associate-*l* 98.9%

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

   7. sub-neg 98.9%

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

   8. metadata-eval 98.9%

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

   9. +-commutative 98.9%

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

   10. mul-1-neg 98.9%

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

   11. sub-neg 98.9%

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

   12. sub-neg 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in uy around inf 98.9%

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

8. Taylor expanded in maxCos around 0 98.5%

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

9. Final simplification 98.5%

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

Alternative 4: 97.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.3%

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

   1. associate-*l* 55.3%

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

   2. sub-neg 55.3%

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

   3. +-commutative 55.3%

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

   4. distribute-rgt-neg-in 55.3%

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

   5. fma-def 55.5%

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

3. Simplified 55.5%

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

4. Taylor expanded in ux around -inf 98.9%

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

   1. +-commutative 98.9%

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

   2. mul-1-neg 98.9%

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

   3. unsub-neg 98.9%

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

   4. *-commutative 98.9%

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

   5. *-commutative 98.9%

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

   6. associate-*l* 98.9%

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

   7. sub-neg 98.9%

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

   8. metadata-eval 98.9%

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

   9. +-commutative 98.9%

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

   10. mul-1-neg 98.9%

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

   11. sub-neg 98.9%

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

   12. sub-neg 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in uy around inf 98.9%

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

8. Taylor expanded in maxCos around 0 98.1%

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

9. Final simplification 98.1%

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

Alternative 5: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 0.00044999999227002263:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 0.00044999999227002263)
   (* (cos (* uy (* 2.0 PI))) (sqrt (- (* ux 2.0) (pow ux 2.0))))
   (* (cos (* PI (* 2.0 uy))) (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 0.00044999999227002263f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((ux * 2.0f) - powf(ux, 2.0f)));
	} else {
		tmp = cosf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(0.00044999999227002263))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(ux * Float32(2.0)) - (ux ^ Float32(2.0)))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(0.00044999999227002263))
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt(((ux * single(2.0)) - (ux ^ single(2.0))));
	else
		tmp = cos((single(pi) * (single(2.0) * uy))) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.49999992e-4

   1. Initial program 56.6%

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

      1. associate-*l* 56.6%

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

      2. sub-neg 56.6%

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

      3. +-commutative 56.6%

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

      4. distribute-rgt-neg-in 56.6%

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

      5. fma-def 56.7%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in ux around 0 98.9%

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

      1. fma-def 98.9%

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

      2. +-commutative 98.9%

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

      3. sub-neg 98.9%

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

      4. metadata-eval 98.9%

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

      5. +-commutative 98.9%

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

      6. distribute-lft-in 98.9%

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

      7. metadata-eval 98.9%

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

      8. associate--l+ 98.9%

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

      9. mul-1-neg 98.9%

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

      10. sub-neg 98.9%

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

      11. *-commutative 98.9%

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

      12. *-commutative 98.9%

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

      13. sub-neg 98.9%

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

      14. mul-1-neg 98.9%

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

      15. associate-*l* 98.9%

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

      16. sub-neg 98.9%

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

      17. metadata-eval 98.9%

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

      18. +-commutative 98.9%

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

      19. mul-1-neg 98.9%

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

      20. sub-neg 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in maxCos around 0 97.2%

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

      1. +-commutative 97.2%

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

      2. mul-1-neg 97.2%

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

      3. unsub-neg 97.2%

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

      4. *-commutative 97.2%

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

   9. Simplified 97.2%

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

   if 4.49999992e-4 < maxCos

   1. Initial program 42.5%

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

   2. Taylor expanded in ux around 0 88.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 0.00044999999227002263:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 6: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00011999999696854502:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux + -2\right) \cdot \left(-ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.00011999999696854502)
   (sqrt
    (+
     (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))
     (* ux (- 2.0 (* 2.0 maxCos)))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* (+ ux -2.0) (- ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.00011999999696854502f) {
		tmp = sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f))) + (ux * (2.0f - (2.0f * maxCos)))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((ux + -2.0f) * -ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.00011999999696854502))
		tmp = sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0)))) + Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(ux + Float32(-2.0)) * Float32(-ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.00011999999696854502))
		tmp = sqrt((((ux ^ single(2.0)) * ((single(1.0) - maxCos) * (maxCos + single(-1.0)))) + (ux * (single(2.0) - (single(2.0) * maxCos)))));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt(((ux + single(-2.0)) * -ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 1.19999997e-4

   1. Initial program 55.2%

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

      1. associate-*l* 55.2%

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

      2. sub-neg 55.2%

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

      3. +-commutative 55.2%

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

      4. distribute-rgt-neg-in 55.2%

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

      5. fma-def 55.4%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in ux around -inf 99.5%

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

      1. +-commutative 99.5%

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

      2. mul-1-neg 99.5%

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

      3. unsub-neg 99.5%

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

      4. *-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*l* 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. +-commutative 99.5%

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

      10. mul-1-neg 99.5%

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

      11. sub-neg 99.5%

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

      12. sub-neg 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in uy around 0 99.4%

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

   if 1.19999997e-4 < (*.f32 uy 2)

   1. Initial program 55.4%

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

      1. associate-*l* 55.4%

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

      2. sub-neg 55.4%

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

      3. +-commutative 55.4%

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

      4. distribute-rgt-neg-in 55.4%

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

      5. fma-def 55.6%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in maxCos around 0 54.6%

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

   5. Taylor expanded in ux around 0 91.6%

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

      1. mul-1-neg 91.6%

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

      2. metadata-eval 91.6%

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

      3. distribute-lft-neg-in 91.6%

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

      4. distribute-neg-out 91.6%

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

      5. unpow2 91.6%

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

      6. distribute-rgt-out 91.6%

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

   7. Simplified 91.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00011999999696854502:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux + -2\right) \cdot \left(-ux\right)}\\ \end{array} \]

Alternative 7: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 0.00044999999227002263:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux + -2\right) \cdot \left(-ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 0.00044999999227002263)
   (* (cos (* uy (* 2.0 PI))) (sqrt (* (+ ux -2.0) (- ux))))
   (* (cos (* PI (* 2.0 uy))) (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 0.00044999999227002263f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((ux + -2.0f) * -ux));
	} else {
		tmp = cosf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(0.00044999999227002263))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(ux + Float32(-2.0)) * Float32(-ux))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(0.00044999999227002263))
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt(((ux + single(-2.0)) * -ux));
	else
		tmp = cos((single(pi) * (single(2.0) * uy))) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.49999992e-4

   1. Initial program 56.6%

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

      1. associate-*l* 56.6%

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

      2. sub-neg 56.6%

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

      3. +-commutative 56.6%

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

      4. distribute-rgt-neg-in 56.6%

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

      5. fma-def 56.7%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in maxCos around 0 56.1%

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

   5. Taylor expanded in ux around 0 97.2%

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

      1. mul-1-neg 97.2%

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

      2. metadata-eval 97.2%

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

      3. distribute-lft-neg-in 97.2%

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

      4. distribute-neg-out 97.2%

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

      5. unpow2 97.2%

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

      6. distribute-rgt-out 97.2%

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

   7. Simplified 97.2%

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

   if 4.49999992e-4 < maxCos

   1. Initial program 42.5%

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

   2. Taylor expanded in ux around 0 88.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 0.00044999999227002263:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux + -2\right) \cdot \left(-ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 8: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.005200000014156103:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.005200000014156103)
   (sqrt
    (+
     (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))
     (* ux (- 2.0 (* 2.0 maxCos)))))
   (* (cos (* PI (* 2.0 uy))) (sqrt (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.005200000014156103f) {
		tmp = sqrtf(((powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f))) + (ux * (2.0f - (2.0f * maxCos)))));
	} else {
		tmp = cosf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.005200000014156103))
		tmp = sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0)))) + Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.005200000014156103))
		tmp = sqrt((((ux ^ single(2.0)) * ((single(1.0) - maxCos) * (maxCos + single(-1.0)))) + (ux * (single(2.0) - (single(2.0) * maxCos)))));
	else
		tmp = cos((single(pi) * (single(2.0) * uy))) * sqrt((ux * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00520000001

   1. Initial program 54.6%

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

      1. associate-*l* 54.6%

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

      2. sub-neg 54.6%

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

      3. +-commutative 54.6%

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

      4. distribute-rgt-neg-in 54.6%

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

      5. fma-def 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in ux around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. unsub-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*l* 99.4%

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

      7. sub-neg 99.4%

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

      8. metadata-eval 99.4%

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

      9. +-commutative 99.4%

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

      10. mul-1-neg 99.4%

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

      11. sub-neg 99.4%

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

      12. sub-neg 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in uy around 0 95.8%

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

   if 0.00520000001 < (*.f32 uy 2)

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around 0 45.1%

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

   3. Taylor expanded in maxCos around 0 71.2%

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

      1. *-commutative 71.2%

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

   5. Simplified 71.2%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.005200000014156103:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]

Alternative 9: 80.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((((ux ** 2.0e0) * ((1.0e0 - maxcos) * (maxcos + (-1.0e0)))) + (ux * (2.0e0 - (2.0e0 * maxcos)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 55.3%

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

   1. associate-*l* 55.3%

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

   2. sub-neg 55.3%

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

   3. +-commutative 55.3%

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

   4. distribute-rgt-neg-in 55.3%

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

   5. fma-def 55.5%

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

3. Simplified 55.5%

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

4. Taylor expanded in ux around -inf 98.9%

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

   1. +-commutative 98.9%

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

   2. mul-1-neg 98.9%

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

   3. unsub-neg 98.9%

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

   4. *-commutative 98.9%

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

   5. *-commutative 98.9%

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

   6. associate-*l* 98.9%

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

   7. sub-neg 98.9%

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

   8. metadata-eval 98.9%

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

   9. +-commutative 98.9%

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

   10. mul-1-neg 98.9%

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

   11. sub-neg 98.9%

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

   12. sub-neg 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in uy around 0 81.1%

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

8. Final simplification 81.1%

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

Alternative 10: 75.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + ux \cdot \left(maxCos + -1\right)\\ \mathbf{if}\;ux \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - t_0 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ 1.0 (* ux (+ maxCos -1.0)))))
   (if (<= ux 9.000000136438757e-5)
     (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
     (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 1.0f + (ux * (maxCos + -1.0f));
	float tmp;
	if (ux <= 9.000000136438757e-5f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f - (t_0 * t_0)));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: t_0
    real(4) :: tmp
    t_0 = 1.0e0 + (ux * (maxcos + (-1.0e0)))
    if (ux <= 9.000000136438757e-5) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt((1.0e0 - (t_0 * t_0)))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(1.0) + Float32(ux * Float32(maxCos + Float32(-1.0))))
	tmp = Float32(0.0)
	if (ux <= Float32(9.000000136438757e-5))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(1.0) + (ux * (maxCos + single(-1.0)));
	tmp = single(0.0);
	if (ux <= single(9.000000136438757e-5))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) - (t_0 * t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 9.00000014e-5

   1. Initial program 34.4%

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

   2. Taylor expanded in ux around 0 93.4%

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

   3. Taylor expanded in uy around 0 77.7%

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

   if 9.00000014e-5 < ux

   1. Initial program 88.5%

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

      1. associate-*l* 88.5%

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

      2. sub-neg 88.5%

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

      3. +-commutative 88.5%

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

      4. distribute-rgt-neg-in 88.5%

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

      5. fma-def 88.7%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in uy around 0 74.3%

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

   5. Taylor expanded in ux around -inf 74.3%

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

      1. mul-1-neg 74.3%

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

      2. unsub-neg 74.3%

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

      3. mul-1-neg 74.3%

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

      4. sub-neg 74.3%

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

   7. Simplified 74.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}\\ \end{array} \]

Alternative 11: 75.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 9.000000136438757e-5)
   (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
   (sqrt
    (+ 1.0 (* (+ 1.0 (* ux (+ maxCos -1.0))) (+ ux (- -1.0 (* ux maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 9.000000136438757e-5f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f + (ux * (maxCos + -1.0f))) * (ux + (-1.0f - (ux * maxCos))))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 9.000000136438757e-5) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 + (ux * (maxcos + (-1.0e0)))) * (ux + ((-1.0e0) - (ux * maxcos))))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(9.000000136438757e-5))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(maxCos + Float32(-1.0)))) * Float32(ux + Float32(Float32(-1.0) - Float32(ux * maxCos))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(9.000000136438757e-5))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) + (ux * (maxCos + single(-1.0)))) * (ux + (single(-1.0) - (ux * maxCos))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 9.00000014e-5

   1. Initial program 34.4%

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

   2. Taylor expanded in ux around 0 93.4%

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

   3. Taylor expanded in uy around 0 77.7%

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

   if 9.00000014e-5 < ux

   1. Initial program 88.5%

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

      1. associate-*l* 88.5%

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

      2. sub-neg 88.5%

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

      3. +-commutative 88.5%

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

      4. distribute-rgt-neg-in 88.5%

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

      5. fma-def 88.7%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in uy around 0 74.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + \left(-1 - ux \cdot maxCos\right)\right)}\\ \end{array} \]

Alternative 12: 73.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 9.000000136438757e-5)
   (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
   (sqrt (+ 1.0 (* (- 1.0 ux) (+ -1.0 (* ux (- 1.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 9.000000136438757e-5f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (-1.0f + (ux * (1.0f - maxCos))))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 9.000000136438757e-5) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * ((-1.0e0) + (ux * (1.0e0 - maxcos))))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(9.000000136438757e-5))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(Float32(-1.0) + Float32(ux * Float32(Float32(1.0) - maxCos))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(9.000000136438757e-5))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (single(-1.0) + (ux * (single(1.0) - maxCos))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 9.00000014e-5

   1. Initial program 34.4%

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

   2. Taylor expanded in ux around 0 93.4%

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

   3. Taylor expanded in uy around 0 77.7%

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

   if 9.00000014e-5 < ux

   1. Initial program 88.5%

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

      1. associate-*l* 88.5%

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

      2. sub-neg 88.5%

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

      3. +-commutative 88.5%

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

      4. distribute-rgt-neg-in 88.5%

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

      5. fma-def 88.7%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in uy around 0 74.3%

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

   5. Taylor expanded in maxCos around 0 72.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]

Alternative 13: 73.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 9.000000136438757e-5)
   (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
   (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 9.000000136438757e-5f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 9.000000136438757e-5) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * (ux + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(9.000000136438757e-5))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(9.000000136438757e-5))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 9.00000014e-5

   1. Initial program 34.4%

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

   2. Taylor expanded in ux around 0 93.4%

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

   3. Taylor expanded in uy around 0 77.7%

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

   if 9.00000014e-5 < ux

   1. Initial program 88.5%

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

      1. associate-*l* 88.5%

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

      2. sub-neg 88.5%

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

      3. +-commutative 88.5%

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

      4. distribute-rgt-neg-in 88.5%

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

      5. fma-def 88.7%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in uy around 0 74.3%

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

   5. Taylor expanded in maxCos around 0 72.2%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 9.000000136438757 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \]

Alternative 14: 64.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 55.3%

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

2. Taylor expanded in ux around 0 77.7%

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

3. Taylor expanded in uy around 0 66.3%

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

4. Final simplification 66.3%

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

Alternative 15: 62.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot 2} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 55.3%

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

2. Taylor expanded in ux around 0 77.7%

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

3. Taylor expanded in uy around 0 66.3%

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

4. Taylor expanded in maxCos around 0 63.3%

   \[\leadsto \sqrt{\color{blue}{2 \cdot ux}} \]
5. Step-by-step derivation

   1. *-commutative 63.3%

      \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

6. Simplified 63.3%

   \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

7. Final simplification 63.3%

   \[\leadsto \sqrt{ux \cdot 2} \]

Reproduce

herbie shell --seed 2023334 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :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)))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))