UniformSampleCone, x

Percentage Accurate: 57.9% → 98.9%

Time: 21.8s

Alternatives: 10

Speedup: 2.2×

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: 57.9% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.5%

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

3. Taylor expanded in ux around 0 57.5%

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

4. Taylor expanded in maxCos around 0 99.0%

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

   1. associate-*r* 99.0%

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

   2. mul-1-neg 99.0%

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

   3. *-commutative 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

6. Simplified 99.0%

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

Alternative 2: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq 0.9997900128364563:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(maxCos \cdot -2 + ux \cdot \left(-1 + maxCos \cdot \left(2 - maxCos\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* (* uy 2.0) PI))))
   (if (<= t_0 0.9997900128364563)
     (* t_0 (sqrt (* ux (- 2.0 ux))))
     (sqrt
      (*
       ux
       (+
        2.0
        (+ (* maxCos -2.0) (* ux (+ -1.0 (* maxCos (- 2.0 maxCos)))))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf(((uy * 2.0f) * ((float) M_PI)));
	float tmp;
	if (t_0 <= 0.9997900128364563f) {
		tmp = t_0 * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf((ux * (2.0f + ((maxCos * -2.0f) + (ux * (-1.0f + (maxCos * (2.0f - maxCos))))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9997900128364563))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(maxCos * Float32(-2.0)) + Float32(ux * Float32(Float32(-1.0) + Float32(maxCos * Float32(Float32(2.0) - maxCos))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos(((uy * single(2.0)) * single(pi)));
	tmp = single(0.0);
	if (t_0 <= single(0.9997900128364563))
		tmp = t_0 * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = sqrt((ux * (single(2.0) + ((maxCos * single(-2.0)) + (ux * (single(-1.0) + (maxCos * (single(2.0) - maxCos))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999790013

   1. Initial program 52.5%

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

   3. Taylor expanded in ux around 0 98.0%

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

   4. Taylor expanded in maxCos around 0 92.4%

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

      1. neg-mul-1 92.4%

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

      2. unsub-neg 92.4%

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

   6. Simplified 92.4%

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

   if 0.999790013 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 55.3%

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

      1. associate-*l* 55.3%

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

      2. sub-neg 55.3%

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

      3. +-commutative 55.3%

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

      4. distribute-rgt-neg-in 55.3%

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

      5. fma-define 55.1%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in uy around 0 54.9%

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

   7. Simplified 54.5%

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

   8. Taylor expanded in ux around inf 94.8%

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

   9. Taylor expanded in maxCos around 0 94.8%

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

       1. associate--l+ 94.8%

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

       2. associate-*r/ 94.8%

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

       3. metadata-eval 94.8%

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

       4. mul-1-neg 94.8%

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

       5. associate-*r/ 94.8%

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

       6. metadata-eval 94.8%

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

   11. Simplified 94.8%

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

   12. Taylor expanded in ux around 0 94.9%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9997900128364563:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(maxCos \cdot -2 + ux \cdot \left(-1 + maxCos \cdot \left(2 - maxCos\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.5%

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

3. Taylor expanded in ux around 0 99.0%

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

4. Final simplification 99.0%

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

Alternative 4: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.5%

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

3. Taylor expanded in ux around 0 57.5%

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

4. Taylor expanded in maxCos around 0 98.0%

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

   1. associate-*r* 98.0%

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

   2. mul-1-neg 98.0%

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

   3. *-commutative 98.0%

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

   4. sub-neg 98.0%

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

   5. metadata-eval 98.0%

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

6. Simplified 98.0%

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

Alternative 5: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0010000000474974513:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(maxCos \cdot -2 + ux \cdot \left(-1 + maxCos \cdot \left(2 - maxCos\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0010000000474974513)
   (sqrt
    (*
     ux
     (+ 2.0 (+ (* maxCos -2.0) (* ux (+ -1.0 (* maxCos (- 2.0 maxCos))))))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0010000000474974513f) {
		tmp = sqrtf((ux * (2.0f + ((maxCos * -2.0f) + (ux * (-1.0f + (maxCos * (2.0f - maxCos))))))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0010000000474974513))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(maxCos * Float32(-2.0)) + Float32(ux * Float32(Float32(-1.0) + Float32(maxCos * Float32(Float32(2.0) - maxCos))))))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0010000000474974513))
		tmp = sqrt((ux * (single(2.0) + ((maxCos * single(-2.0)) + (ux * (single(-1.0) + (maxCos * (single(2.0) - maxCos))))))));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00100000005

   1. Initial program 55.4%

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

      1. associate-*l* 55.4%

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

      2. sub-neg 55.4%

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

      3. +-commutative 55.4%

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

      4. distribute-rgt-neg-in 55.4%

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

      5. fma-define 55.2%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in uy around 0 55.3%

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

   7. Simplified 54.9%

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

   8. Taylor expanded in ux around inf 97.0%

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

   9. Taylor expanded in maxCos around 0 97.0%

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

       1. associate--l+ 97.0%

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

       2. associate-*r/ 97.0%

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

       3. metadata-eval 97.0%

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

       4. mul-1-neg 97.0%

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

       5. associate-*r/ 97.0%

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

       6. metadata-eval 97.0%

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

   11. Simplified 97.0%

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

   12. Taylor expanded in ux around 0 97.1%

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

   if 0.00100000005 < uy

   1. Initial program 52.8%

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

      1. associate-*l* 52.8%

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

      2. sub-neg 52.8%

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

      3. +-commutative 52.8%

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

      4. distribute-rgt-neg-in 52.8%

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

      5. fma-define 53.1%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in maxCos around 0 50.0%

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

   6. Taylor expanded in ux around 0 75.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0010000000474974513:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(maxCos \cdot -2 + ux \cdot \left(-1 + maxCos \cdot \left(2 - maxCos\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. sub-neg 54.5%

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

   3. +-commutative 54.5%

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

   4. distribute-rgt-neg-in 54.5%

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

   5. fma-define 54.4%

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

3. Simplified 54.8%

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

5. Taylor expanded in uy around 0 46.2%

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

7. Simplified 46.0%

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

8. Taylor expanded in ux around inf 76.6%

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

9. Taylor expanded in maxCos around 0 76.6%

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

    1. associate--l+ 76.6%

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

    2. associate-*r/ 76.6%

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

    3. metadata-eval 76.6%

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

    4. mul-1-neg 76.6%

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

    5. associate-*r/ 76.6%

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

    6. metadata-eval 76.6%

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

11. Simplified 76.6%

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

12. Taylor expanded in ux around 0 76.8%

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

13. Final simplification 76.8%

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

Alternative 7: 75.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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) - (ux * ux)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. sub-neg 54.5%

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

   3. +-commutative 54.5%

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

   4. distribute-rgt-neg-in 54.5%

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

   5. fma-define 54.4%

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

3. Simplified 54.8%

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

5. Taylor expanded in uy around 0 46.2%

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

7. Simplified 46.0%

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

8. Taylor expanded in ux around inf 76.6%

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

9. Taylor expanded in maxCos around 0 70.7%

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

    1. sub-neg 70.7%

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

    2. associate-*r/ 70.7%

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

    3. metadata-eval 70.7%

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

    4. metadata-eval 70.7%

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

11. Simplified 70.7%

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

12. Taylor expanded in ux around 0 70.8%

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

    1. mul-1-neg 70.8%

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

    2. unsub-neg 70.8%

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

14. Simplified 70.8%

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

    1. sub-neg 70.8%

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

    2. distribute-rgt-in 70.9%

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

16. Applied egg-rr 70.9%

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

17. Final simplification 70.9%

    \[\leadsto \sqrt{2 \cdot ux - ux \cdot ux} \]
18. Add Preprocessing

Alternative 8: 75.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. sub-neg 54.5%

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

   3. +-commutative 54.5%

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

   4. distribute-rgt-neg-in 54.5%

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

   5. fma-define 54.4%

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

3. Simplified 54.8%

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

5. Taylor expanded in uy around 0 46.2%

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

7. Simplified 46.0%

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

8. Taylor expanded in ux around inf 76.6%

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

9. Taylor expanded in maxCos around 0 70.7%

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

    1. sub-neg 70.7%

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

    2. associate-*r/ 70.7%

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

    3. metadata-eval 70.7%

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

    4. metadata-eval 70.7%

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

11. Simplified 70.7%

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

12. Taylor expanded in ux around 0 70.8%

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

    1. mul-1-neg 70.8%

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

    2. unsub-neg 70.8%

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

14. Simplified 70.8%

    \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 - ux\right)}} \]
15. Add Preprocessing

Alternative 9: 61.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

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))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. sub-neg 54.5%

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

   3. +-commutative 54.5%

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

   4. distribute-rgt-neg-in 54.5%

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

   5. fma-define 54.4%

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

3. Simplified 54.8%

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

5. Taylor expanded in uy around 0 46.2%

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

7. Simplified 46.0%

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

8. Taylor expanded in ux around 0 63.5%

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

9. Taylor expanded in maxCos around 0 60.0%

   \[\leadsto \sqrt{\color{blue}{2 \cdot ux}} \]
10. Add Preprocessing

Alternative 10: 20.0% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 1.0)

C

float code(float ux, float uy, float maxCos) {
	return 1.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(1.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0);
end

Derivation

1. Initial program 54.5%

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

   1. associate-*l* 54.5%

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

   2. sub-neg 54.5%

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

   3. +-commutative 54.5%

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

   4. distribute-rgt-neg-in 54.5%

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

   5. fma-define 54.4%

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

3. Simplified 54.8%

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

5. Taylor expanded in uy around 0 46.2%

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

7. Simplified 46.0%

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

8. Taylor expanded in ux around -inf 19.1%

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

   1. mul-1-neg 19.1%

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

   2. associate-*r* 19.1%

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

   3. distribute-rgt-neg-in 19.1%

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

   4. sub-neg 19.1%

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

   5. metadata-eval 19.1%

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

   6. distribute-neg-in 19.1%

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

   7. mul-1-neg 19.1%

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

   8. metadata-eval 19.1%

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

   9. +-commutative 19.1%

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

   10. associate-*r* 19.1%

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

   11. unpow2 19.1%

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

   12. swap-sqr 19.1%

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

   13. +-commutative 19.1%

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

   14. fma-undefine 19.1%

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

   15. +-commutative 19.1%

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

   16. fma-undefine 19.1%

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

   17. unpow2 19.1%

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

   18. fma-undefine 19.1%

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

   19. +-commutative 19.1%

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

   20. mul-1-neg 19.1%

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

   21. unsub-neg 19.1%

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

10. Simplified 19.1%

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

11. Taylor expanded in ux around 0 19.2%

    \[\leadsto \color{blue}{1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))))))