UniformSampleCone, x

Percentage Accurate: 57.0% → 98.9%

Time: 23.6s

Alternatives: 19

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: 57.0% 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: 98.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in ux around 0 98.8%

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

   1. fma-def 98.8%

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

   2. +-commutative 98.8%

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

   3. sub-neg 98.8%

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

   4. metadata-eval 98.8%

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

   5. +-commutative 98.8%

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

   6. distribute-lft-in 98.8%

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

   7. metadata-eval 98.8%

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

   8. associate--l+ 98.8%

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

   9. mul-1-neg 98.8%

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

   10. sub-neg 98.8%

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

   11. *-commutative 98.8%

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

   12. sub-neg 98.8%

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

   13. metadata-eval 98.8%

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

6. Simplified 98.8%

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

   1. add-cbrt-cube 98.8%

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

   2. add-cbrt-cube 98.8%

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

   3. cbrt-unprod 98.8%

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

8. Applied egg-rr 98.9%

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

10. Simplified 98.9%

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

11. Final simplification 98.9%

    \[\leadsto \sqrt[3]{{\cos \left(-2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(maxCos + -1\right)}^{2}\right)}^{1.5}} \]

Alternative 2: 98.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in ux around 0 98.8%

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

   1. fma-def 98.8%

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

   2. +-commutative 98.8%

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

   3. sub-neg 98.8%

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

   4. metadata-eval 98.8%

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

   5. +-commutative 98.8%

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

   6. distribute-lft-in 98.8%

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

   7. metadata-eval 98.8%

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

   8. associate--l+ 98.8%

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

   9. mul-1-neg 98.8%

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

   10. sub-neg 98.8%

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

   11. *-commutative 98.8%

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

   12. sub-neg 98.8%

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

   13. metadata-eval 98.8%

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

6. Simplified 98.8%

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

   1. add-cbrt-cube 98.8%

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

   2. add-cbrt-cube 98.8%

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

   3. cbrt-unprod 98.8%

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

8. Applied egg-rr 98.9%

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

10. Simplified 98.9%

    \[\leadsto \color{blue}{\sqrt[3]{{\cos \left(-2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \left(2 + maxCos \cdot -2\right) - {ux}^{2} \cdot {\left(maxCos + -1\right)}^{2}\right)}^{1.5}}} \]
11. Step-by-step derivation

    1. expm1-log1p-u 98.9%

       \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(-2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \left(2 + maxCos \cdot -2\right) - {ux}^{2} \cdot {\left(maxCos + -1\right)}^{2}\right)}^{1.5}\right)\right)}} \]

    2. expm1-udef 40.8%

       \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left({\cos \left(-2 \cdot \left(uy \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \left(2 + maxCos \cdot -2\right) - {ux}^{2} \cdot {\left(maxCos + -1\right)}^{2}\right)}^{1.5}\right)} - 1}} \]

12. Applied egg-rr 40.8%

    \[\leadsto \sqrt[3]{\color{blue}{e^{\mathsf{log1p}\left({\cos \left(\left(-2 \cdot uy\right) \cdot \pi\right)}^{3} \cdot {\left(ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)}^{1.5}\right)} - 1}} \]
13. Step-by-step derivation

    1. expm1-def 98.9%

       \[\leadsto \sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\left(-2 \cdot uy\right) \cdot \pi\right)}^{3} \cdot {\left(ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}\right)}^{1.5}\right)\right)}} \]

    2. expm1-log1p 98.9%

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

    3. *-commutative 98.9%

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

    4. *-commutative 98.9%

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

14. Simplified 98.9%

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

15. Final simplification 98.9%

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

Alternative 3: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in ux around 0 98.8%

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

   1. fma-def 98.8%

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

   2. +-commutative 98.8%

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

   3. sub-neg 98.8%

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

   4. metadata-eval 98.8%

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

   5. +-commutative 98.8%

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

   6. distribute-lft-in 98.8%

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

   7. metadata-eval 98.8%

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

   8. associate--l+ 98.8%

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

   9. mul-1-neg 98.8%

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

   10. sub-neg 98.8%

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

   11. *-commutative 98.8%

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

   12. sub-neg 98.8%

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

   13. metadata-eval 98.8%

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

6. Simplified 98.8%

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

7. Taylor expanded in uy around inf 98.8%

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

   1. *-commutative 98.8%

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

   2. sub-neg 98.8%

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

   3. metadata-eval 98.8%

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

   4. distribute-rgt-neg-in 98.8%

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

   5. mul-1-neg 98.8%

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

   6. distribute-lft-in 98.8%

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

   7. mul-1-neg 98.8%

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

   8. sub-neg 98.8%

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

   9. sub-neg 98.8%

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

   10. metadata-eval 98.8%

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

   11. fma-udef 98.8%

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

9. Simplified 98.8%

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

10. Final simplification 98.8%

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

Alternative 4: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in ux around -inf 98.8%

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

   1. +-commutative 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

   4. *-commutative 98.8%

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

   5. mul-1-neg 98.8%

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

   6. sub-neg 98.8%

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

   7. sub-neg 98.8%

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

   8. metadata-eval 98.8%

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

   9. sub-neg 98.8%

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

   10. mul-1-neg 98.8%

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

   11. unsub-neg 98.8%

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

   12. mul-1-neg 98.8%

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

   13. sub-neg 98.8%

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

   14. metadata-eval 98.8%

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

6. Simplified 98.8%

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

7. Taylor expanded in uy around inf 98.8%

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

8. Final simplification 98.8%

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

Alternative 5: 96.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 1.99999995e-4

   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 58.2%

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

   4. Taylor expanded in 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 99.2%

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

      1. fma-def 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. metadata-eval 99.2%

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

      4. sub-neg 99.2%

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

      5. metadata-eval 99.2%

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

   9. Simplified 99.2%

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

   if 1.99999995e-4 < (*.f32 uy 2)

   1. Initial program 57.7%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in ux around 0 98.0%

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

      1. fma-def 98.0%

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

      2. +-commutative 98.0%

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

      3. sub-neg 98.0%

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

      4. metadata-eval 98.0%

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

      5. +-commutative 98.0%

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

      6. distribute-lft-in 98.0%

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

      7. metadata-eval 98.0%

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

      8. associate--l+ 98.1%

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

      9. mul-1-neg 98.1%

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

      10. sub-neg 98.1%

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

      11. *-commutative 98.1%

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

      12. sub-neg 98.1%

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

      13. metadata-eval 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in maxCos around 0 93.1%

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

      1. +-commutative 93.1%

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

      2. neg-mul-1 93.1%

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

      3. unsub-neg 93.1%

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

      4. *-commutative 93.1%

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

   9. Simplified 93.1%

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

4. Final simplification 96.5%

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

Alternative 6: 97.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in ux around 0 98.8%

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

   1. fma-def 98.8%

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

   2. +-commutative 98.8%

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

   3. sub-neg 98.8%

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

   4. metadata-eval 98.8%

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

   5. +-commutative 98.8%

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

   6. distribute-lft-in 98.8%

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

   7. metadata-eval 98.8%

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

   8. associate--l+ 98.8%

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

   9. mul-1-neg 98.8%

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

   10. sub-neg 98.8%

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

   11. *-commutative 98.8%

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

   12. sub-neg 98.8%

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

   13. metadata-eval 98.8%

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

6. Simplified 98.8%

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

   1. add-cbrt-cube 98.8%

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

   2. pow1/3 95.6%

      \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \color{blue}{{\left(\left(\sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}\right)}^{0.3333333333333333}} \]

8. Applied egg-rr 95.6%

   \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

9. Taylor expanded in maxCos around 0 94.5%

   \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot {\left({\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), \color{blue}{-1 \cdot {ux}^{2}}\right)\right)}^{1.5}\right)}^{0.3333333333333333} \]
10. Step-by-step derivation

    1. neg-mul-1 94.5%

       \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot {\left({\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), \color{blue}{-{ux}^{2}}\right)\right)}^{1.5}\right)}^{0.3333333333333333} \]

11. Simplified 94.5%

    \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot {\left({\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), \color{blue}{-{ux}^{2}}\right)\right)}^{1.5}\right)}^{0.3333333333333333} \]
12. Step-by-step derivation

    1. expm1-log1p-u 94.5%

       \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), -{ux}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)\right)} \]

    2. expm1-udef 80.6%

       \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), -{ux}^{2}\right)\right)}^{1.5}\right)}^{0.3333333333333333}\right)} - 1\right)} \]

    3. pow-pow 80.7%

       \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), -{ux}^{2}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} - 1\right) \]

    4. metadata-eval 80.7%

       \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \left(e^{\mathsf{log1p}\left({\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), -{ux}^{2}\right)\right)}^{\color{blue}{0.5}}\right)} - 1\right) \]

    5. pow1/2 80.7%

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

13. Applied egg-rr 80.7%

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

    1. expm1-def 97.5%

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

    2. expm1-log1p 97.7%

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

    3. fma-udef 97.6%

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

    4. unsub-neg 97.6%

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

15. Simplified 97.6%

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

16. Final simplification 97.6%

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

Alternative 7: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + -2 \cdot maxCos, {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot maxCos\right) + ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0024999999441206455)
   (sqrt
    (fma
     ux
     (+ 2.0 (* -2.0 maxCos))
     (* (pow ux 2.0) (* (+ maxCos -1.0) (- 1.0 maxCos)))))
   (* (cos (* PI (* uy 2.0))) (sqrt (+ (* -2.0 (* ux maxCos)) (* ux 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0024999999441206455f) {
		tmp = sqrtf(fmaf(ux, (2.0f + (-2.0f * maxCos)), (powf(ux, 2.0f) * ((maxCos + -1.0f) * (1.0f - maxCos)))));
	} else {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf(((-2.0f * (ux * maxCos)) + (ux * 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0024999999441206455))
		tmp = 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)))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(Float32(Float32(-2.0) * Float32(ux * maxCos)) + Float32(ux * Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00249999994

   1. Initial program 57.6%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in ux around 0 99.3%

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

      1. fma-def 99.3%

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

      2. +-commutative 99.3%

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

      3. sub-neg 99.3%

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

      4. metadata-eval 99.3%

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

      5. +-commutative 99.3%

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

      6. distribute-lft-in 99.3%

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

      7. metadata-eval 99.3%

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

      8. associate--l+ 99.3%

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

      9. mul-1-neg 99.3%

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

      10. sub-neg 99.3%

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

      11. *-commutative 99.3%

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

      12. sub-neg 99.3%

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

      13. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in uy around 0 96.9%

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

      1. fma-def 96.9%

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

      2. cancel-sign-sub-inv 96.9%

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

      3. metadata-eval 96.9%

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

      4. sub-neg 96.9%

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

      5. metadata-eval 96.9%

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

   9. Simplified 96.9%

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

   if 0.00249999994 < (*.f32 uy 2)

   1. Initial program 58.5%

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

   2. Taylor expanded in ux around 0 49.6%

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

   3. Taylor expanded in maxCos around 0 77.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + -2 \cdot maxCos, {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot maxCos\right) + ux \cdot 2}\\ \end{array} \]

Alternative 8: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot maxCos\right) + ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0024999999441206455)
   (sqrt
    (+
     (* (pow ux 2.0) (* (+ maxCos -1.0) (- 1.0 maxCos)))
     (* ux (- 2.0 (* 2.0 maxCos)))))
   (* (cos (* PI (* uy 2.0))) (sqrt (+ (* -2.0 (* ux maxCos)) (* ux 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0024999999441206455f) {
		tmp = sqrtf(((powf(ux, 2.0f) * ((maxCos + -1.0f) * (1.0f - maxCos))) + (ux * (2.0f - (2.0f * maxCos)))));
	} else {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf(((-2.0f * (ux * maxCos)) + (ux * 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0024999999441206455))
		tmp = sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))) + Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(Float32(Float32(-2.0) * Float32(ux * maxCos)) + Float32(ux * Float32(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0024999999441206455))
		tmp = sqrt((((ux ^ single(2.0)) * ((maxCos + single(-1.0)) * (single(1.0) - maxCos))) + (ux * (single(2.0) - (single(2.0) * maxCos)))));
	else
		tmp = cos((single(pi) * (uy * single(2.0)))) * sqrt(((single(-2.0) * (ux * maxCos)) + (ux * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00249999994

   1. Initial program 57.6%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in ux around -inf 99.3%

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

      1. +-commutative 99.3%

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

      2. mul-1-neg 99.3%

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

      3. unsub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. mul-1-neg 99.3%

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

      6. sub-neg 99.3%

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

      7. sub-neg 99.3%

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

      8. metadata-eval 99.3%

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

      9. sub-neg 99.3%

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

      10. mul-1-neg 99.3%

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

      11. unsub-neg 99.3%

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

      12. mul-1-neg 99.3%

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

      13. sub-neg 99.3%

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

      14. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in uy around 0 96.9%

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

   if 0.00249999994 < (*.f32 uy 2)

   1. Initial program 58.5%

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

   2. Taylor expanded in ux around 0 49.6%

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

   3. Taylor expanded in maxCos around 0 77.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{-2 \cdot \left(ux \cdot maxCos\right) + ux \cdot 2}\\ \end{array} \]

Alternative 9: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(2 - 2 \cdot maxCos\right)\\ \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) + t_0}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 2.0 (* 2.0 maxCos)))))
   (if (<= (* uy 2.0) 0.0024999999441206455)
     (sqrt (+ (* (pow ux 2.0) (* (+ maxCos -1.0) (- 1.0 maxCos))) t_0))
     (* (cos (* PI (* uy 2.0))) (sqrt t_0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (2.0f - (2.0f * maxCos));
	float tmp;
	if ((uy * 2.0f) <= 0.0024999999441206455f) {
		tmp = sqrtf(((powf(ux, 2.0f) * ((maxCos + -1.0f) * (1.0f - maxCos))) + t_0));
	} else {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf(t_0);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0024999999441206455))
		tmp = sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))) + t_0));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = ux * (single(2.0) - (single(2.0) * maxCos));
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0024999999441206455))
		tmp = sqrt((((ux ^ single(2.0)) * ((maxCos + single(-1.0)) * (single(1.0) - maxCos))) + t_0));
	else
		tmp = cos((single(pi) * (uy * single(2.0)))) * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00249999994

   1. Initial program 57.6%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in ux around -inf 99.3%

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

      1. +-commutative 99.3%

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

      2. mul-1-neg 99.3%

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

      3. unsub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. mul-1-neg 99.3%

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

      6. sub-neg 99.3%

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

      7. sub-neg 99.3%

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

      8. metadata-eval 99.3%

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

      9. sub-neg 99.3%

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

      10. mul-1-neg 99.3%

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

      11. unsub-neg 99.3%

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

      12. mul-1-neg 99.3%

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

      13. sub-neg 99.3%

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

      14. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in uy around 0 96.9%

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

   if 0.00249999994 < (*.f32 uy 2)

   1. Initial program 58.5%

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

   2. Taylor expanded in ux around 0 77.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 10: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0024999999441206455)
   (sqrt
    (+
     (* (pow ux 2.0) (* (+ maxCos -1.0) (- 1.0 maxCos)))
     (* ux (- 2.0 (* 2.0 maxCos)))))
   (* (cos (* PI (* uy 2.0))) (sqrt (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0024999999441206455f) {
		tmp = sqrtf(((powf(ux, 2.0f) * ((maxCos + -1.0f) * (1.0f - maxCos))) + (ux * (2.0f - (2.0f * maxCos)))));
	} else {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0024999999441206455))
		tmp = sqrt(Float32(Float32((ux ^ Float32(2.0)) * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))) + Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(ux * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.0024999999441206455))
		tmp = sqrt((((ux ^ single(2.0)) * ((maxCos + single(-1.0)) * (single(1.0) - maxCos))) + (ux * (single(2.0) - (single(2.0) * maxCos)))));
	else
		tmp = cos((single(pi) * (uy * single(2.0)))) * sqrt((ux * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00249999994

   1. Initial program 57.6%

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

   3. Simplified 57.7%

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

   4. Taylor expanded in ux around -inf 99.3%

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

      1. +-commutative 99.3%

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

      2. mul-1-neg 99.3%

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

      3. unsub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. mul-1-neg 99.3%

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

      6. sub-neg 99.3%

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

      7. sub-neg 99.3%

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

      8. metadata-eval 99.3%

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

      9. sub-neg 99.3%

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

      10. mul-1-neg 99.3%

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

      11. unsub-neg 99.3%

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

      12. mul-1-neg 99.3%

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

      13. sub-neg 99.3%

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

      14. metadata-eval 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in uy around 0 96.9%

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

   if 0.00249999994 < (*.f32 uy 2)

   1. Initial program 58.5%

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

   2. Taylor expanded in ux around 0 49.6%

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

   3. Taylor expanded in maxCos around 0 74.6%

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

      1. *-commutative 74.6%

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

   5. Simplified 74.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{{ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) + ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]

Alternative 11: 79.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in ux around -inf 98.8%

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

   1. +-commutative 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

   4. *-commutative 98.8%

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

   5. mul-1-neg 98.8%

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

   6. sub-neg 98.8%

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

   7. sub-neg 98.8%

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

   8. metadata-eval 98.8%

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

   9. sub-neg 98.8%

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

   10. mul-1-neg 98.8%

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

   11. unsub-neg 98.8%

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

   12. mul-1-neg 98.8%

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

   13. sub-neg 98.8%

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

   14. metadata-eval 98.8%

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

6. Simplified 98.8%

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

7. Taylor expanded in uy around 0 78.3%

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

8. Final simplification 78.3%

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

Alternative 12: 78.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in ux around 0 98.8%

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

   1. fma-def 98.8%

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

   2. +-commutative 98.8%

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

   3. sub-neg 98.8%

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

   4. metadata-eval 98.8%

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

   5. +-commutative 98.8%

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

   6. distribute-lft-in 98.8%

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

   7. metadata-eval 98.8%

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

   8. associate--l+ 98.8%

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

   9. mul-1-neg 98.8%

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

   10. sub-neg 98.8%

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

   11. *-commutative 98.8%

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

   12. sub-neg 98.8%

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

   13. metadata-eval 98.8%

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

6. Simplified 98.8%

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

   1. add-cbrt-cube 98.8%

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

   2. pow1/3 95.6%

      \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \color{blue}{{\left(\left(\sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(1 - maxCos\right) + \left(1 - maxCos\right), {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}\right)}^{0.3333333333333333}} \]

8. Applied egg-rr 95.6%

   \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

9. Taylor expanded in maxCos around 0 94.5%

   \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot {\left({\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), \color{blue}{-1 \cdot {ux}^{2}}\right)\right)}^{1.5}\right)}^{0.3333333333333333} \]
10. Step-by-step derivation

    1. neg-mul-1 94.5%

       \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot {\left({\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), \color{blue}{-{ux}^{2}}\right)\right)}^{1.5}\right)}^{0.3333333333333333} \]

11. Simplified 94.5%

    \[\leadsto \cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot {\left({\left(\mathsf{fma}\left(ux, 2 \cdot \left(1 - maxCos\right), \color{blue}{-{ux}^{2}}\right)\right)}^{1.5}\right)}^{0.3333333333333333} \]

12. Taylor expanded in uy around 0 77.8%

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

13. Final simplification 77.8%

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

Alternative 13: 75.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in ux around 0 98.8%

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

   1. fma-def 98.8%

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

   2. +-commutative 98.8%

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

   3. sub-neg 98.8%

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

   4. metadata-eval 98.8%

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

   5. +-commutative 98.8%

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

   6. distribute-lft-in 98.8%

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

   7. metadata-eval 98.8%

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

   8. associate--l+ 98.8%

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

   9. mul-1-neg 98.8%

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

   10. sub-neg 98.8%

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

   11. *-commutative 98.8%

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

   12. sub-neg 98.8%

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

   13. metadata-eval 98.8%

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

6. Simplified 98.8%

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

7. Taylor expanded in uy around 0 78.3%

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

   1. fma-def 78.3%

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

   2. cancel-sign-sub-inv 78.3%

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

   3. metadata-eval 78.3%

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

   4. sub-neg 78.3%

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

   5. metadata-eval 78.3%

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

9. Simplified 78.3%

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

10. Taylor expanded in maxCos around 0 74.2%

    \[\leadsto \color{blue}{\sqrt{-1 \cdot {ux}^{2} + 2 \cdot ux}} \]
11. Step-by-step derivation

    1. +-commutative 74.2%

       \[\leadsto \sqrt{\color{blue}{2 \cdot ux + -1 \cdot {ux}^{2}}} \]

    2. neg-mul-1 74.2%

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

    3. sub-neg 74.2%

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

12. Simplified 74.2%

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

13. Final simplification 74.2%

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

Alternative 14: 74.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + ux \cdot \left(maxCos + -1\right)\\ \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot 2 + ux \cdot \left(-2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - t_0 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ 1.0 (* ux (+ maxCos -1.0)))))
   (if (<= ux 0.00011000000085914508)
     (sqrt (+ (* ux 2.0) (* ux (* -2.0 maxCos))))
     (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = 1.0f + (ux * (maxCos + -1.0f));
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf(((ux * 2.0f) + (ux * (-2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f - (t_0 * t_0)));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: t_0
    real(4) :: tmp
    t_0 = 1.0e0 + (ux * (maxcos + (-1.0e0)))
    if (ux <= 0.00011000000085914508e0) then
        tmp = sqrt(((ux * 2.0e0) + (ux * ((-2.0e0) * maxcos))))
    else
        tmp = sqrt((1.0e0 - (t_0 * t_0)))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(1.0) + Float32(ux * Float32(maxCos + Float32(-1.0))))
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(Float32(ux * Float32(2.0)) + Float32(ux * Float32(Float32(-2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(1.0) + (ux * (maxCos + single(-1.0)));
	tmp = single(0.0);
	if (ux <= single(0.00011000000085914508))
		tmp = sqrt(((ux * single(2.0)) + (ux * (single(-2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) - (t_0 * t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.10000001e-4

   1. Initial program 39.3%

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

   3. Simplified 39.2%

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

   4. Taylor expanded in ux around 0 98.7%

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

      1. fma-def 98.7%

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

      2. +-commutative 98.7%

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

      3. sub-neg 98.7%

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

      4. metadata-eval 98.7%

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

      5. +-commutative 98.7%

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

      6. distribute-lft-in 98.7%

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

      7. metadata-eval 98.7%

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

      8. associate--l+ 98.8%

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

      9. mul-1-neg 98.8%

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

      10. sub-neg 98.8%

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

      11. *-commutative 98.8%

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

      12. sub-neg 98.8%

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

      13. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in uy around 0 77.7%

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

      1. fma-def 77.7%

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

      2. cancel-sign-sub-inv 77.7%

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

      3. metadata-eval 77.7%

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

      4. sub-neg 77.7%

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

      5. metadata-eval 77.7%

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

   9. Simplified 77.7%

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

   10. Taylor expanded in ux around 0 73.9%

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

       1. distribute-lft-in 73.9%

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

   12. Applied egg-rr 73.9%

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

   if 1.10000001e-4 < ux

   1. Initial program 88.4%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in uy around 0 72.1%

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

   5. Taylor expanded in ux around -inf 72.3%

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

      1. mul-1-neg 72.3%

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

      2. unsub-neg 72.3%

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

      3. mul-1-neg 72.3%

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

      4. sub-neg 72.3%

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

   7. Simplified 72.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\sqrt{ux \cdot 2 + ux \cdot \left(-2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}\\ \end{array} \]

Alternative 15: 73.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0005000000237487257:\\ \;\;\;\;\sqrt{ux \cdot 2 + ux \cdot \left(-2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0005000000237487257)
   (sqrt (+ (* ux 2.0) (* ux (* -2.0 maxCos))))
   (sqrt (+ 1.0 (* (+ 1.0 (* ux (+ maxCos -1.0))) (+ ux -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0005000000237487257f) {
		tmp = sqrtf(((ux * 2.0f) + (ux * (-2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f + (ux * (maxCos + -1.0f))) * (ux + -1.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.0005000000237487257e0) then
        tmp = sqrt(((ux * 2.0e0) + (ux * ((-2.0e0) * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 + (ux * (maxcos + (-1.0e0)))) * (ux + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0005000000237487257))
		tmp = sqrt(Float32(Float32(ux * Float32(2.0)) + Float32(ux * Float32(Float32(-2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(maxCos + Float32(-1.0)))) * Float32(ux + Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.0005000000237487257))
		tmp = sqrt(((ux * single(2.0)) + (ux * (single(-2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) + (ux * (maxCos + single(-1.0)))) * (ux + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 5.00000024e-4

   1. Initial program 42.3%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in ux around 0 98.8%

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

      1. fma-def 98.8%

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

      2. +-commutative 98.8%

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

      3. sub-neg 98.8%

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

      4. metadata-eval 98.8%

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

      5. +-commutative 98.8%

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

      6. distribute-lft-in 98.8%

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

      7. metadata-eval 98.8%

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

      8. associate--l+ 98.8%

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

      9. mul-1-neg 98.8%

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

      10. sub-neg 98.8%

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

      11. *-commutative 98.8%

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

      12. sub-neg 98.8%

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

      13. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in uy around 0 78.3%

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

      1. fma-def 78.3%

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

      2. cancel-sign-sub-inv 78.3%

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

      3. metadata-eval 78.3%

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

      4. sub-neg 78.3%

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

      5. metadata-eval 78.3%

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

   9. Simplified 78.3%

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

   10. Taylor expanded in ux around 0 73.1%

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

       1. distribute-lft-in 73.1%

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

   12. Applied egg-rr 73.1%

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

   if 5.00000024e-4 < ux

   1. Initial program 91.0%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in uy around 0 73.3%

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

   5. Taylor expanded in maxCos around 0 71.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0005000000237487257:\\ \;\;\;\;\sqrt{ux \cdot 2 + ux \cdot \left(-2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + -1\right)}\\ \end{array} \]

Alternative 16: 73.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0005000000237487257:\\ \;\;\;\;\sqrt{ux \cdot 2 + ux \cdot \left(-2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0005000000237487257)
   (sqrt (+ (* ux 2.0) (* ux (* -2.0 maxCos))))
   (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0005000000237487257f) {
		tmp = sqrtf(((ux * 2.0f) + (ux * (-2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.0005000000237487257e0) then
        tmp = sqrt(((ux * 2.0e0) + (ux * ((-2.0e0) * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * (ux + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0005000000237487257))
		tmp = sqrt(Float32(Float32(ux * Float32(2.0)) + Float32(ux * Float32(Float32(-2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.0005000000237487257))
		tmp = sqrt(((ux * single(2.0)) + (ux * (single(-2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 5.00000024e-4

   1. Initial program 42.3%

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

   3. Simplified 42.2%

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

   4. Taylor expanded in ux around 0 98.8%

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

      1. fma-def 98.8%

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

      2. +-commutative 98.8%

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

      3. sub-neg 98.8%

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

      4. metadata-eval 98.8%

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

      5. +-commutative 98.8%

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

      6. distribute-lft-in 98.8%

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

      7. metadata-eval 98.8%

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

      8. associate--l+ 98.8%

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

      9. mul-1-neg 98.8%

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

      10. sub-neg 98.8%

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

      11. *-commutative 98.8%

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

      12. sub-neg 98.8%

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

      13. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in uy around 0 78.3%

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

      1. fma-def 78.3%

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

      2. cancel-sign-sub-inv 78.3%

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

      3. metadata-eval 78.3%

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

      4. sub-neg 78.3%

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

      5. metadata-eval 78.3%

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

   9. Simplified 78.3%

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

   10. Taylor expanded in ux around 0 73.1%

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

       1. distribute-lft-in 73.1%

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

   12. Applied egg-rr 73.1%

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

   if 5.00000024e-4 < ux

   1. Initial program 91.0%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in uy around 0 73.3%

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

   5. Taylor expanded in maxCos around 0 70.9%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0005000000237487257:\\ \;\;\;\;\sqrt{ux \cdot 2 + ux \cdot \left(-2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \]

Alternative 17: 64.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in ux around 0 98.8%

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

   1. fma-def 98.8%

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

   2. +-commutative 98.8%

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

   3. sub-neg 98.8%

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

   4. metadata-eval 98.8%

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

   5. +-commutative 98.8%

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

   6. distribute-lft-in 98.8%

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

   7. metadata-eval 98.8%

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

   8. associate--l+ 98.8%

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

   9. mul-1-neg 98.8%

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

   10. sub-neg 98.8%

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

   11. *-commutative 98.8%

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

   12. sub-neg 98.8%

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

   13. metadata-eval 98.8%

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

6. Simplified 98.8%

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

7. Taylor expanded in uy around 0 78.3%

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

   1. fma-def 78.3%

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

   2. cancel-sign-sub-inv 78.3%

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

   3. metadata-eval 78.3%

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

   4. sub-neg 78.3%

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

   5. metadata-eval 78.3%

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

9. Simplified 78.3%

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

10. Taylor expanded in ux around 0 64.2%

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

    1. distribute-lft-in 64.2%

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

12. Applied egg-rr 64.2%

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

13. Final simplification 64.2%

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

Alternative 18: 64.6% 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 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in ux around 0 98.8%

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

   1. fma-def 98.8%

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

   2. +-commutative 98.8%

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

   3. sub-neg 98.8%

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

   4. metadata-eval 98.8%

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

   5. +-commutative 98.8%

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

   6. distribute-lft-in 98.8%

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

   7. metadata-eval 98.8%

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

   8. associate--l+ 98.8%

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

   9. mul-1-neg 98.8%

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

   10. sub-neg 98.8%

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

   11. *-commutative 98.8%

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

   12. sub-neg 98.8%

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

   13. metadata-eval 98.8%

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

6. Simplified 98.8%

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

7. Taylor expanded in uy around 0 78.3%

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

   1. fma-def 78.3%

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

   2. cancel-sign-sub-inv 78.3%

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

   3. metadata-eval 78.3%

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

   4. sub-neg 78.3%

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

   5. metadata-eval 78.3%

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

9. Simplified 78.3%

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

10. Taylor expanded in ux around 0 64.2%

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

11. Final simplification 64.2%

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

Alternative 19: 61.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

3. Simplified 57.9%

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

4. Taylor expanded in ux around 0 98.8%

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

   1. fma-def 98.8%

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

   2. +-commutative 98.8%

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

   3. sub-neg 98.8%

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

   4. metadata-eval 98.8%

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

   5. +-commutative 98.8%

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

   6. distribute-lft-in 98.8%

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

   7. metadata-eval 98.8%

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

   8. associate--l+ 98.8%

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

   9. mul-1-neg 98.8%

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

   10. sub-neg 98.8%

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

   11. *-commutative 98.8%

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

   12. sub-neg 98.8%

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

   13. metadata-eval 98.8%

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

6. Simplified 98.8%

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

7. Taylor expanded in uy around 0 78.3%

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

   1. fma-def 78.3%

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

   2. cancel-sign-sub-inv 78.3%

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

   3. metadata-eval 78.3%

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

   4. sub-neg 78.3%

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

   5. metadata-eval 78.3%

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

9. Simplified 78.3%

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

10. Taylor expanded in ux around 0 64.2%

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

11. Taylor expanded in maxCos around 0 61.5%

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

12. Final simplification 61.5%

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

Reproduce

herbie shell --seed 2023298 
(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)))))))