UniformSampleCone, x

Percentage Accurate: 57.4% → 99.0%

Time: 18.2s

Alternatives: 16

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: 57.4% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.4%

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

3. Simplified 55.7%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. fma-def 99.0%

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

   2. +-commutative 99.0%

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

   3. sub-neg 99.0%

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

   4. metadata-eval 99.0%

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

   5. +-commutative 99.0%

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

   6. distribute-lft-in 99.0%

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

   7. metadata-eval 99.0%

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

   8. associate--l+ 99.0%

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

   9. mul-1-neg 99.0%

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

   10. sub-neg 99.0%

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

   11. *-commutative 99.0%

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

   12. sub-neg 99.0%

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

   13. metadata-eval 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in uy around inf 99.0%

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

   1. *-commutative 99.0%

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

   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. *-commutative 99.0%

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

   4. sub-neg 99.0%

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

   5. 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%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. expm1-log1p-u 55.3%

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

   2. expm1-udef 55.2%

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

   3. log1p-udef 55.0%

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

   4. add-exp-log 55.0%

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

   5. pow2 55.0%

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

   6. +-commutative 55.0%

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

   7. associate-+r- 54.8%

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

   8. fma-udef 54.8%

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

3. Applied egg-rr 54.8%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. cancel-sign-sub-inv 99.0%

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

   3. metadata-eval 99.0%

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

   4. mul-1-neg 99.0%

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

   5. unsub-neg 99.0%

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

   6. *-commutative 99.0%

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

   7. unpow2 99.0%

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

   8. unpow2 99.0%

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

   9. swap-sqr 99.0%

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

   10. sub-neg 99.0%

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

   11. metadata-eval 99.0%

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

   12. sub-neg 99.0%

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

   13. metadata-eval 99.0%

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

   14. unpow1 99.0%

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

   15. pow-plus 99.0%

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

   16. metadata-eval 99.0%

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

6. Simplified 99.0%

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

7. Final simplification 99.0%

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

Alternative 3: 95.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\cos \left(-2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.999999987376214e-7)
   (* (cos (* -2.0 (* uy 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 <= 4.999999987376214e-7f) {
		tmp = cosf((-2.0f * (uy * ((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(4.999999987376214e-7))
		tmp = Float32(cos(Float32(Float32(-2.0) * Float32(uy * 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(4.999999987376214e-7))
		tmp = cos((single(-2.0) * (uy * 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 < 4.99999999e-7

   1. Initial program 58.0%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in ux around 0 99.0%

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

      1. fma-def 99.0%

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

      2. +-commutative 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

      5. +-commutative 99.0%

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

      6. distribute-lft-in 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate--l+ 99.0%

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

      9. mul-1-neg 99.0%

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

      10. sub-neg 99.0%

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

      11. *-commutative 99.0%

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

      12. sub-neg 99.0%

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

      13. metadata-eval 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in maxCos around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. +-commutative 98.9%

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

      3. mul-1-neg 98.9%

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

      4. unsub-neg 98.9%

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

      5. *-commutative 98.9%

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

   9. Simplified 98.9%

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

   if 4.99999999e-7 < maxCos

   1. Initial program 34.8%

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

   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 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\cos \left(-2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 4: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9998775124549866:\\ \;\;\;\;\cos \left(uy \cdot \left(-2 \cdot \pi\right)\right) \cdot \sqrt{1 + \left(-1 - ux \cdot \left(ux + -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (+ (- 1.0 ux) (* ux maxCos)) 0.9998775124549866)
   (* (cos (* uy (* -2.0 PI))) (sqrt (+ 1.0 (- -1.0 (* ux (+ 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 (((1.0f - ux) + (ux * maxCos)) <= 0.9998775124549866f) {
		tmp = cosf((uy * (-2.0f * ((float) M_PI)))) * sqrtf((1.0f + (-1.0f - (ux * (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 (Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos)) <= Float32(0.9998775124549866))
		tmp = Float32(cos(Float32(uy * Float32(Float32(-2.0) * Float32(pi)))) * sqrt(Float32(Float32(1.0) + Float32(Float32(-1.0) - Float32(ux * Float32(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 (((single(1.0) - ux) + (ux * maxCos)) <= single(0.9998775124549866))
		tmp = cos((uy * (single(-2.0) * single(pi)))) * sqrt((single(1.0) + (single(-1.0) - (ux * (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 (+.f32 (-.f32 1 ux) (*.f32 ux maxCos)) < 0.999877512

   1. Initial program 89.2%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in maxCos around 0 88.7%

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

   5. Taylor expanded in ux around 0 90.5%

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

      1. unpow2 90.5%

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

      2. distribute-rgt-out 90.5%

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

   7. Simplified 90.5%

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

   if 0.999877512 < (+.f32 (-.f32 1 ux) (*.f32 ux maxCos))

   1. Initial program 33.4%

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

   2. Taylor expanded in ux around 0 93.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9998775124549866:\\ \;\;\;\;\cos \left(uy \cdot \left(-2 \cdot \pi\right)\right) \cdot \sqrt{1 + \left(-1 - ux \cdot \left(ux + -2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 5: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - 2 \cdot maxCos\\ \mathbf{if}\;2 \cdot uy \leq 0.0012000000569969416:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, t_0, {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- 2.0 (* 2.0 maxCos))))
   (if (<= (* 2.0 uy) 0.0012000000569969416)
     (sqrt (fma ux t_0 (* (pow ux 2.0) (* (+ maxCos -1.0) (- 1.0 maxCos)))))
     (* (cos (* PI (* 2.0 uy))) (sqrt (* ux t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 2.0f - (2.0f * maxCos);
	float tmp;
	if ((2.0f * uy) <= 0.0012000000569969416f) {
		tmp = sqrtf(fmaf(ux, t_0, (powf(ux, 2.0f) * ((maxCos + -1.0f) * (1.0f - maxCos)))));
	} else {
		tmp = cosf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * t_0));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0012000000569969416))
		tmp = sqrt(fma(ux, t_0, Float32((ux ^ Float32(2.0)) * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.2%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in ux around 0 99.4%

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

      1. fma-def 99.4%

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

      2. +-commutative 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate--l+ 99.4%

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

      9. mul-1-neg 99.4%

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

      10. sub-neg 99.4%

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

      11. *-commutative 99.4%

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

      12. sub-neg 99.4%

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

      13. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in uy around 0 97.9%

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

      1. fma-def 97.9%

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

      2. *-commutative 97.9%

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

      3. sub-neg 97.9%

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

      4. metadata-eval 97.9%

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

   9. Simplified 97.9%

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

   if 0.00120000006 < (*.f32 uy 2)

   1. Initial program 57.9%

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

   2. Taylor expanded in ux around 0 76.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0012000000569969416:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 - 2 \cdot maxCos, {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 6: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0012000000569969416:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + maxCos \cdot -2\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0012000000569969416)
   (sqrt (- (* ux (+ 2.0 (* maxCos -2.0))) (pow (* ux (+ maxCos -1.0)) 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 ((2.0f * uy) <= 0.0012000000569969416f) {
		tmp = sqrtf(((ux * (2.0f + (maxCos * -2.0f))) - powf((ux * (maxCos + -1.0f)), 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 (Float32(Float32(2.0) * uy) <= Float32(0.0012000000569969416))
		tmp = sqrt(Float32(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))) - (Float32(ux * Float32(maxCos + Float32(-1.0))) ^ 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 ((single(2.0) * uy) <= single(0.0012000000569969416))
		tmp = sqrt(((ux * (single(2.0) + (maxCos * single(-2.0)))) - ((ux * (maxCos + single(-1.0))) ^ 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 (*.f32 uy 2) < 0.00120000006

   1. Initial program 54.2%

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

      1. expm1-log1p-u 54.1%

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

      2. expm1-udef 54.0%

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

      3. log1p-udef 53.8%

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

      4. add-exp-log 53.8%

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

      5. pow2 53.8%

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

      6. +-commutative 53.8%

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

      7. associate-+r- 53.6%

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

      8. fma-udef 53.6%

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

   3. Applied egg-rr 53.6%

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

   4. Taylor expanded in ux around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. cancel-sign-sub-inv 99.4%

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

      3. metadata-eval 99.4%

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

      4. mul-1-neg 99.4%

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

      5. unsub-neg 99.4%

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

      6. *-commutative 99.4%

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

      7. unpow2 99.4%

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

      8. unpow2 99.4%

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

      9. swap-sqr 99.4%

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

      10. sub-neg 99.4%

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

      11. metadata-eval 99.4%

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

      12. sub-neg 99.4%

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

      13. metadata-eval 99.4%

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

      14. unpow1 99.4%

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

      15. pow-plus 99.4%

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

      16. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in uy around 0 97.9%

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

   if 0.00120000006 < (*.f32 uy 2)

   1. Initial program 57.9%

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

   2. Taylor expanded in ux around 0 76.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0012000000569969416:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + maxCos \cdot -2\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 7: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0031999999191612005:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + maxCos \cdot -2\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(-2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0031999999191612005)
   (sqrt (- (* ux (+ 2.0 (* maxCos -2.0))) (pow (* ux (+ maxCos -1.0)) 2.0)))
   (* (cos (* uy (* -2.0 PI))) (sqrt (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0031999999191612005f) {
		tmp = sqrtf(((ux * (2.0f + (maxCos * -2.0f))) - powf((ux * (maxCos + -1.0f)), 2.0f)));
	} else {
		tmp = cosf((uy * (-2.0f * ((float) M_PI)))) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0031999999191612005))
		tmp = sqrt(Float32(Float32(ux * Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0)))) - (Float32(ux * Float32(maxCos + Float32(-1.0))) ^ Float32(2.0))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(-2.0) * Float32(pi)))) * 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(0.0031999999191612005))
		tmp = sqrt(((ux * (single(2.0) + (maxCos * single(-2.0)))) - ((ux * (maxCos + single(-1.0))) ^ single(2.0))));
	else
		tmp = cos((uy * (single(-2.0) * single(pi)))) * sqrt((ux * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00319999992

   1. Initial program 54.2%

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

      1. expm1-log1p-u 54.1%

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

      2. expm1-udef 54.0%

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

      3. log1p-udef 53.8%

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

      4. add-exp-log 53.8%

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

      5. pow2 53.8%

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

      6. +-commutative 53.8%

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

      7. associate-+r- 53.7%

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

      8. fma-udef 53.7%

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

   3. Applied egg-rr 53.7%

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

   4. Taylor expanded in ux around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. cancel-sign-sub-inv 99.4%

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

      3. metadata-eval 99.4%

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

      4. mul-1-neg 99.4%

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

      5. unsub-neg 99.4%

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

      6. *-commutative 99.4%

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

      7. unpow2 99.4%

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

      8. unpow2 99.4%

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

      9. swap-sqr 99.4%

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

      10. sub-neg 99.4%

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

      11. metadata-eval 99.4%

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

      12. sub-neg 99.4%

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

      13. metadata-eval 99.4%

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

      14. unpow1 99.4%

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

      15. pow-plus 99.4%

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

      16. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in uy around 0 96.5%

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

   if 0.00319999992 < (*.f32 uy 2)

   1. Initial program 58.5%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in maxCos around 0 57.2%

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

   5. Taylor expanded in ux around 0 72.8%

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

      1. *-commutative 72.8%

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

   7. Simplified 72.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0031999999191612005:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + maxCos \cdot -2\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(-2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]

Alternative 8: 79.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((ux * (2.0f + (maxCos * -2.0f))) - powf((ux * (maxCos + -1.0f)), 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 + (maxcos * (-2.0e0)))) - ((ux * (maxcos + (-1.0e0))) ** 2.0e0)))
end function

Julia

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

MATLAB

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

Derivation

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. expm1-log1p-u 55.3%

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

   2. expm1-udef 55.2%

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

   3. log1p-udef 55.0%

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

   4. add-exp-log 55.0%

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

   5. pow2 55.0%

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

   6. +-commutative 55.0%

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

   7. associate-+r- 54.8%

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

   8. fma-udef 54.8%

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

3. Applied egg-rr 54.8%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. cancel-sign-sub-inv 99.0%

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

   3. metadata-eval 99.0%

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

   4. mul-1-neg 99.0%

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

   5. unsub-neg 99.0%

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

   6. *-commutative 99.0%

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

   7. unpow2 99.0%

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

   8. unpow2 99.0%

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

   9. swap-sqr 99.0%

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

   10. sub-neg 99.0%

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

   11. metadata-eval 99.0%

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

   12. sub-neg 99.0%

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

   13. metadata-eval 99.0%

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

   14. unpow1 99.0%

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

   15. pow-plus 99.0%

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

   16. metadata-eval 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in uy around 0 80.2%

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

8. Final simplification 80.2%

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

Alternative 9: 73.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00014000000373926014:\\ \;\;\;\;\sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(-1 - \left({ux}^{2} + ux \cdot -2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00014000000373926014)
   (sqrt (- (* ux (- -2.0)) (* 2.0 (* ux maxCos))))
   (sqrt (+ 1.0 (- -1.0 (+ (pow ux 2.0) (* ux -2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00014000000373926014f) {
		tmp = sqrtf(((ux * -(-2.0f)) - (2.0f * (ux * maxCos))));
	} else {
		tmp = sqrtf((1.0f + (-1.0f - (powf(ux, 2.0f) + (ux * -2.0f)))));
	}
	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 <= 0.00014000000373926014e0) then
        tmp = sqrt(((ux * -(-2.0e0)) - (2.0e0 * (ux * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((-1.0e0) - ((ux ** 2.0e0) + (ux * (-2.0e0))))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00014000000373926014))
		tmp = sqrt(Float32(Float32(ux * Float32(-Float32(-2.0))) - Float32(Float32(2.0) * Float32(ux * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(-1.0) - Float32((ux ^ Float32(2.0)) + Float32(ux * Float32(-2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00014000000373926014))
		tmp = sqrt(((ux * -single(-2.0)) - (single(2.0) * (ux * maxCos))));
	else
		tmp = sqrt((single(1.0) + (single(-1.0) - ((ux ^ single(2.0)) + (ux * single(-2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.40000004e-4

   1. Initial program 33.9%

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

   3. Simplified 33.9%

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

   4. Taylor expanded in uy around 0 29.9%

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

   5. Taylor expanded in ux around 0 76.8%

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

   6. Taylor expanded in maxCos around 0 76.8%

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

   if 1.40000004e-4 < ux

   1. Initial program 89.6%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in uy around 0 75.2%

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

   5. Taylor expanded in ux around -inf 76.3%

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

   6. Taylor expanded in maxCos around 0 75.6%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00014000000373926014:\\ \;\;\;\;\sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(-1 - \left({ux}^{2} + ux \cdot -2\right)\right)}\\ \end{array} \]

Alternative 10: 74.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00013000000035390258:\\ \;\;\;\;\sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(\left(1 + ux \cdot maxCos\right) - ux\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00013000000035390258)
   (sqrt (- (* ux (- -2.0)) (* 2.0 (* ux maxCos))))
   (sqrt
    (+ 1.0 (* (- (+ 1.0 (* ux maxCos)) ux) (+ -1.0 (* ux (- 1.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00013000000035390258f) {
		tmp = sqrtf(((ux * -(-2.0f)) - (2.0f * (ux * maxCos))));
	} else {
		tmp = sqrtf((1.0f + (((1.0f + (ux * maxCos)) - 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 <= 0.00013000000035390258e0) then
        tmp = sqrt(((ux * -(-2.0e0)) - (2.0e0 * (ux * maxcos))))
    else
        tmp = sqrt((1.0e0 + (((1.0e0 + (ux * maxcos)) - 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(0.00013000000035390258))
		tmp = sqrt(Float32(Float32(ux * Float32(-Float32(-2.0))) - Float32(Float32(2.0) * Float32(ux * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) + Float32(ux * maxCos)) - ux) * Float32(Float32(-1.0) + Float32(ux * Float32(Float32(1.0) - maxCos))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00013000000035390258))
		tmp = sqrt(((ux * -single(-2.0)) - (single(2.0) * (ux * maxCos))));
	else
		tmp = sqrt((single(1.0) + (((single(1.0) + (ux * maxCos)) - ux) * (single(-1.0) + (ux * (single(1.0) - maxCos))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.3e-4

   1. Initial program 33.7%

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

   3. Simplified 33.7%

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

   4. Taylor expanded in uy around 0 29.7%

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

   5. Taylor expanded in ux around 0 76.8%

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

   6. Taylor expanded in maxCos around 0 76.9%

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

   if 1.3e-4 < ux

   1. Initial program 89.4%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in uy around 0 75.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00013000000035390258:\\ \;\;\;\;\sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(\left(1 + ux \cdot maxCos\right) - ux\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]

Alternative 11: 74.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00013000000035390258:\\ \;\;\;\;\sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(ux - \left(1 + ux \cdot maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00013000000035390258)
   (sqrt (- (* ux (- -2.0)) (* 2.0 (* ux maxCos))))
   (sqrt
    (+ 1.0 (* (+ 1.0 (- (* ux maxCos) ux)) (- ux (+ 1.0 (* ux maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00013000000035390258f) {
		tmp = sqrtf(((ux * -(-2.0f)) - (2.0f * (ux * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f + ((ux * maxCos) - ux)) * (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 <= 0.00013000000035390258e0) then
        tmp = sqrt(((ux * -(-2.0e0)) - (2.0e0 * (ux * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 + ((ux * maxcos) - ux)) * (ux - (1.0e0 + (ux * maxcos))))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00013000000035390258))
		tmp = sqrt(Float32(Float32(ux * Float32(-Float32(-2.0))) - Float32(Float32(2.0) * Float32(ux * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(Float32(ux * maxCos) - ux)) * Float32(ux - Float32(Float32(1.0) + Float32(ux * maxCos))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00013000000035390258))
		tmp = sqrt(((ux * -single(-2.0)) - (single(2.0) * (ux * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) + ((ux * maxCos) - ux)) * (ux - (single(1.0) + (ux * maxCos))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.3e-4

   1. Initial program 33.7%

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

   3. Simplified 33.7%

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

   4. Taylor expanded in uy around 0 29.7%

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

   5. Taylor expanded in ux around 0 76.8%

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

   6. Taylor expanded in maxCos around 0 76.9%

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

   if 1.3e-4 < ux

   1. Initial program 89.4%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in uy around 0 75.2%

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

      1. sub-neg 75.2%

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

      2. metadata-eval 75.2%

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

      3. distribute-rgt-in 75.3%

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

      4. *-commutative 75.3%

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

      5. fma-def 75.3%

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

      6. mul-1-neg 75.3%

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

      7. fma-neg 75.3%

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

   6. Applied egg-rr 75.3%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00013000000035390258:\\ \;\;\;\;\sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(ux - \left(1 + ux \cdot maxCos\right)\right)}\\ \end{array} \]

Alternative 12: 73.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00013000000035390258:\\ \;\;\;\;\sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux - \left(1 + ux \cdot maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00013000000035390258)
   (sqrt (- (* ux (- -2.0)) (* 2.0 (* ux maxCos))))
   (sqrt (+ 1.0 (* (- 1.0 ux) (- ux (+ 1.0 (* ux maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00013000000035390258f) {
		tmp = sqrtf(((ux * -(-2.0f)) - (2.0f * (ux * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (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 <= 0.00013000000035390258e0) then
        tmp = sqrt(((ux * -(-2.0e0)) - (2.0e0 * (ux * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * (ux - (1.0e0 + (ux * maxcos))))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00013000000035390258))
		tmp = sqrt(Float32(Float32(ux * Float32(-Float32(-2.0))) - Float32(Float32(2.0) * Float32(ux * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux - Float32(Float32(1.0) + Float32(ux * maxCos))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00013000000035390258))
		tmp = sqrt(((ux * -single(-2.0)) - (single(2.0) * (ux * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (ux - (single(1.0) + (ux * maxCos))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.3e-4

   1. Initial program 33.7%

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

   3. Simplified 33.7%

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

   4. Taylor expanded in uy around 0 29.7%

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

   5. Taylor expanded in ux around 0 76.8%

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

   6. Taylor expanded in maxCos around 0 76.9%

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

   if 1.3e-4 < ux

   1. Initial program 89.4%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in uy around 0 75.2%

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

   5. Taylor expanded in maxCos around 0 74.7%

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

      1. neg-mul-1 74.7%

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

   7. Simplified 74.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00013000000035390258:\\ \;\;\;\;\sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux - \left(1 + ux \cdot maxCos\right)\right)}\\ \end{array} \]

Alternative 13: 73.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00013000000035390258:\\ \;\;\;\;\sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00013000000035390258)
   (sqrt (- (* ux (- -2.0)) (* 2.0 (* ux maxCos))))
   (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00013000000035390258f) {
		tmp = sqrtf(((ux * -(-2.0f)) - (2.0f * (ux * maxCos))));
	} else {
		tmp = sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	}
	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 <= 0.00013000000035390258e0) then
        tmp = sqrt(((ux * -(-2.0e0)) - (2.0e0 * (ux * maxcos))))
    else
        tmp = sqrt((1.0e0 - ((1.0e0 - ux) * (1.0e0 - ux))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00013000000035390258))
		tmp = sqrt(Float32(Float32(ux * Float32(-Float32(-2.0))) - Float32(Float32(2.0) * Float32(ux * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00013000000035390258))
		tmp = sqrt(((ux * -single(-2.0)) - (single(2.0) * (ux * maxCos))));
	else
		tmp = sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.3e-4

   1. Initial program 33.7%

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

   3. Simplified 33.7%

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

   4. Taylor expanded in uy around 0 29.7%

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

   5. Taylor expanded in ux around 0 76.8%

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

   6. Taylor expanded in maxCos around 0 76.9%

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

   if 1.3e-4 < ux

   1. Initial program 89.4%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in uy around 0 75.2%

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

   5. Taylor expanded in maxCos around 0 74.6%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00013000000035390258:\\ \;\;\;\;\sqrt{ux \cdot \left(--2\right) - 2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \]

Alternative 14: 64.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((ux * -(-2.0f)) - (2.0f * (ux * 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 * (ux * maxcos))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 55.4%

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

3. Simplified 55.7%

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

4. Taylor expanded in uy around 0 47.5%

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

5. Taylor expanded in ux around 0 65.3%

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

6. Taylor expanded in maxCos around 0 65.3%

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

7. Final simplification 65.3%

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

Alternative 15: 64.2% 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 55.4%

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

3. Simplified 55.7%

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

4. Taylor expanded in uy around 0 47.5%

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

5. Taylor expanded in ux around 0 65.3%

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

6. Final simplification 65.3%

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

Alternative 16: 61.8% accurate, 3.1× speedup

Math

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

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(-Float32(-2.0))))
end

MATLAB

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

Derivation

1. Initial program 55.4%

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

3. Simplified 55.7%

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

4. Taylor expanded in uy around 0 47.5%

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

5. Taylor expanded in ux around 0 65.3%

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

6. Taylor expanded in maxCos around 0 62.9%

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

   1. *-commutative 62.9%

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

8. Simplified 62.9%

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

9. Final simplification 62.9%

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

Reproduce

herbie shell --seed 2023320 
(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)))))))