UniformSampleCone, x

Percentage Accurate: 57.4% → 99.0%

Time: 17.1s

Alternatives: 16

Speedup: 3.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 53.1%

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

3. Simplified 53.2%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. fma-def 99.2%

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

   2. +-commutative 99.2%

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

   3. sub-neg 99.2%

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

   4. metadata-eval 99.2%

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

   5. +-commutative 99.2%

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

   6. distribute-lft-in 99.2%

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

   7. metadata-eval 99.2%

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

   8. associate--l+ 99.2%

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

   9. mul-1-neg 99.2%

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

   10. sub-neg 99.2%

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

   11. *-commutative 99.2%

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

   12. sub-neg 99.2%

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

   13. metadata-eval 99.2%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

Alternative 2: 96.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999974966049194:\\ \;\;\;\;\cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + -2 \cdot maxCos, {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* PI (* uy 2.0))) 0.9999974966049194)
   (* (cos (* (* PI -2.0) uy)) (sqrt (- (* ux 2.0) (pow ux 2.0))))
   (sqrt
    (fma
     ux
     (+ 2.0 (* -2.0 maxCos))
     (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf((((float) M_PI) * (uy * 2.0f))) <= 0.9999974966049194f) {
		tmp = cosf(((((float) M_PI) * -2.0f) * uy)) * sqrtf(((ux * 2.0f) - powf(ux, 2.0f)));
	} else {
		tmp = sqrtf(fmaf(ux, (2.0f + (-2.0f * maxCos)), (powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) <= Float32(0.9999974966049194))
		tmp = Float32(cos(Float32(Float32(Float32(pi) * Float32(-2.0)) * uy)) * sqrt(Float32(Float32(ux * Float32(2.0)) - (ux ^ Float32(2.0)))));
	else
		tmp = sqrt(fma(ux, Float32(Float32(2.0) + Float32(Float32(-2.0) * maxCos)), Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) < 0.999997497

   1. Initial program 57.4%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in ux around 0 98.5%

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

      1. fma-def 98.5%

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

      2. +-commutative 98.5%

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

      3. sub-neg 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

      6. distribute-lft-in 98.5%

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

      7. metadata-eval 98.5%

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

      8. associate--l+ 98.4%

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

      9. mul-1-neg 98.4%

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

      10. sub-neg 98.4%

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

      11. *-commutative 98.4%

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

      12. sub-neg 98.4%

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

      13. metadata-eval 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in maxCos around 0 95.7%

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

      1. +-commutative 95.7%

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

      2. mul-1-neg 95.7%

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

      3. unsub-neg 95.7%

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

      4. *-commutative 95.7%

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

   9. Simplified 95.7%

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

   if 0.999997497 < (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32)))

   1. Initial program 51.0%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in ux around 0 99.5%

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

      1. fma-def 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-lft-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate--l+ 99.6%

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

      9. mul-1-neg 99.6%

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

      10. sub-neg 99.6%

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

      11. *-commutative 99.6%

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

      12. sub-neg 99.6%

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

      13. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in uy around 0 98.9%

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

      1. fma-def 98.9%

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

      2. cancel-sign-sub-inv 98.9%

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

      3. metadata-eval 98.9%

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

      4. *-commutative 98.9%

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

      5. sub-neg 98.9%

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

      6. mul-1-neg 98.9%

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

      7. mul-1-neg 98.9%

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

      8. sub-neg 98.9%

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

      9. sub-neg 98.9%

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

      10. metadata-eval 98.9%

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

   9. Simplified 98.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999974966049194:\\ \;\;\;\;\cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \sqrt{ux \cdot 2 - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + -2 \cdot maxCos, {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \end{array} \]

Alternative 3: 89.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999964237213135:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot ux\right) \cdot \left(maxCos + -1\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* PI (* uy 2.0)))))
   (if (<= t_0 0.9999964237213135)
     (* t_0 (sqrt (* ux 2.0)))
     (sqrt
      (- (* (* -2.0 ux) (+ maxCos -1.0)) (pow (* ux (+ maxCos -1.0)) 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((((float) M_PI) * (uy * 2.0f)));
	float tmp;
	if (t_0 <= 0.9999964237213135f) {
		tmp = t_0 * sqrtf((ux * 2.0f));
	} else {
		tmp = sqrtf((((-2.0f * ux) * (maxCos + -1.0f)) - powf((ux * (maxCos + -1.0f)), 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(pi) * Float32(uy * Float32(2.0))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999964237213135))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(2.0))));
	else
		tmp = sqrt(Float32(Float32(Float32(Float32(-2.0) * ux) * Float32(maxCos + Float32(-1.0))) - (Float32(ux * Float32(maxCos + Float32(-1.0))) ^ Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos((single(pi) * (uy * single(2.0))));
	tmp = single(0.0);
	if (t_0 <= single(0.9999964237213135))
		tmp = t_0 * sqrt((ux * single(2.0)));
	else
		tmp = sqrt((((single(-2.0) * ux) * (maxCos + single(-1.0))) - ((ux * (maxCos + single(-1.0))) ^ single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) < 0.999996424

   1. Initial program 57.4%

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

   2. Taylor expanded in ux around 0 78.2%

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

   3. Taylor expanded in maxCos around 0 76.7%

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

      1. *-commutative 76.7%

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

   5. Simplified 76.7%

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

      1. expm1-log1p-u 76.7%

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

      2. expm1-udef 65.0%

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

      3. associate-*r* 65.0%

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

      4. pow1/2 65.0%

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

      5. pow1/2 65.0%

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

      6. pow-prod-down 65.0%

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

      7. *-commutative 65.0%

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

      8. *-commutative 65.0%

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

      9. associate-*r* 65.0%

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

   7. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(ux \cdot 2\right)}^{0.5} \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 76.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(ux \cdot 2\right)}^{0.5} \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\right)\right)} \]

      2. expm1-log1p 76.7%

         \[\leadsto \color{blue}{{\left(ux \cdot 2\right)}^{0.5} \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)} \]

      3. unpow1/2 76.7%

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

      4. *-commutative 76.7%

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

   9. Simplified 76.7%

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

   if 0.999996424 < (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32)))

   1. Initial program 51.0%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in uy around 0 51.1%

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

      1. associate--l+ 51.0%

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

      2. fma-neg 51.0%

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

      3. mul-1-neg 51.0%

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

      4. fma-def 51.0%

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

      5. distribute-rgt-in 51.0%

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

      6. metadata-eval 51.0%

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

      7. sub-neg 51.0%

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

      8. +-commutative 51.0%

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

      9. sub-neg 51.0%

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

      10. metadata-eval 51.0%

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

      11. *-commutative 51.0%

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

      12. fma-def 51.0%

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

   6. Applied egg-rr 51.0%

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

   7. Taylor expanded in ux around -inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. unsub-neg 98.8%

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

      3. associate-*r* 98.8%

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

      4. *-commutative 98.8%

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

      5. sub-neg 98.8%

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

      6. metadata-eval 98.8%

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

      7. +-commutative 98.8%

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

      8. unpow2 98.8%

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

      9. unpow2 98.8%

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

      10. swap-sqr 98.8%

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

      11. sub-neg 98.8%

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

      12. metadata-eval 98.8%

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

      13. +-commutative 98.8%

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

      14. sub-neg 98.8%

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

      15. metadata-eval 98.8%

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

      16. +-commutative 98.8%

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

      17. unpow1 98.8%

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

      18. pow-plus 98.8%

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

      19. metadata-eval 98.8%

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

   9. Simplified 98.8%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999964237213135:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot ux\right) \cdot \left(maxCos + -1\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\\ \end{array} \]

Alternative 4: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.1%

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

3. Simplified 53.2%

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

4. Taylor expanded in ux around -inf 99.2%

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

   1. +-commutative 99.2%

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

   2. mul-1-neg 99.2%

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

   3. unsub-neg 99.2%

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

   4. *-commutative 99.2%

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

   5. mul-1-neg 99.2%

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

   6. sub-neg 99.2%

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

   7. sub-neg 99.2%

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

   8. metadata-eval 99.2%

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

   9. sub-neg 99.2%

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

   10. mul-1-neg 99.2%

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

   11. unsub-neg 99.2%

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

   12. mul-1-neg 99.2%

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

   13. sub-neg 99.2%

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

   14. metadata-eval 99.2%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

Alternative 5: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.1%

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

3. Simplified 53.2%

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

4. Taylor expanded in ux around 0 99.2%

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

5. Taylor expanded in maxCos around 0 98.9%

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

6. Final simplification 98.9%

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

Alternative 6: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.1%

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

3. Simplified 53.2%

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

4. Taylor expanded in maxCos around 0 52.9%

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

5. Taylor expanded in ux around 0 98.9%

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

6. Final simplification 98.9%

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

Alternative 7: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008549999911338091:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + -2 \cdot maxCos, {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0008549999911338091)
   (sqrt
    (fma
     ux
     (+ 2.0 (* -2.0 maxCos))
     (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))))
   (* (cos (* PI (* uy 2.0))) (sqrt (* ux (- 2.0 (* maxCos 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0008549999911338091f) {
		tmp = sqrtf(fmaf(ux, (2.0f + (-2.0f * maxCos)), (powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f)))));
	} else {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf((ux * (2.0f - (maxCos * 2.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0008549999911338091))
		tmp = sqrt(fma(ux, Float32(Float32(2.0) + Float32(Float32(-2.0) * maxCos)), Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(maxCos * Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 8.54999991e-4

   1. Initial program 51.3%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in ux around 0 99.5%

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

      1. fma-def 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-lft-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate--l+ 99.6%

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

      9. mul-1-neg 99.6%

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

      10. sub-neg 99.6%

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

      11. *-commutative 99.6%

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

      12. sub-neg 99.6%

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

      13. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in uy around 0 98.7%

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

      1. fma-def 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. metadata-eval 98.7%

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

      4. *-commutative 98.7%

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

      5. sub-neg 98.7%

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

      6. mul-1-neg 98.7%

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

      7. mul-1-neg 98.7%

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

      8. sub-neg 98.7%

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

      9. sub-neg 98.7%

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

      10. metadata-eval 98.7%

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

   9. Simplified 98.7%

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

   if 8.54999991e-4 < (*.f32 uy 2)

   1. Initial program 56.9%

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

   2. Taylor expanded in ux around 0 78.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008549999911338091:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + -2 \cdot maxCos, {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \end{array} \]

Alternative 8: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008549999911338091:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + -2 \cdot maxCos, {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0008549999911338091)
   (sqrt
    (fma
     ux
     (+ 2.0 (* -2.0 maxCos))
     (* (pow ux 2.0) (* (- 1.0 maxCos) (+ maxCos -1.0)))))
   (*
    (cos (* (* PI -2.0) uy))
    (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0008549999911338091f) {
		tmp = sqrtf(fmaf(ux, (2.0f + (-2.0f * maxCos)), (powf(ux, 2.0f) * ((1.0f - maxCos) * (maxCos + -1.0f)))));
	} else {
		tmp = cosf(((((float) M_PI) * -2.0f) * uy)) * sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0008549999911338091))
		tmp = sqrt(fma(ux, Float32(Float32(2.0) + Float32(Float32(-2.0) * maxCos)), Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))));
	else
		tmp = Float32(cos(Float32(Float32(Float32(pi) * Float32(-2.0)) * uy)) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 8.54999991e-4

   1. Initial program 51.3%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in ux around 0 99.5%

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

      1. fma-def 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-lft-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate--l+ 99.6%

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

      9. mul-1-neg 99.6%

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

      10. sub-neg 99.6%

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

      11. *-commutative 99.6%

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

      12. sub-neg 99.6%

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

      13. metadata-eval 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in uy around 0 98.7%

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

      1. fma-def 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. metadata-eval 98.7%

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

      4. *-commutative 98.7%

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

      5. sub-neg 98.7%

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

      6. mul-1-neg 98.7%

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

      7. mul-1-neg 98.7%

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

      8. sub-neg 98.7%

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

      9. sub-neg 98.7%

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

      10. metadata-eval 98.7%

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

   9. Simplified 98.7%

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

   if 8.54999991e-4 < (*.f32 uy 2)

   1. Initial program 56.9%

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

   3. Simplified 56.7%

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

   4. Taylor expanded in ux around 0 78.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008549999911338091:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, 2 + -2 \cdot maxCos, {ux}^{2} \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\pi \cdot -2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \end{array} \]

Alternative 9: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008549999911338091:\\ \;\;\;\;\sqrt{\left(-2 \cdot ux\right) \cdot \left(maxCos + -1\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0008549999911338091)
   (sqrt (- (* (* -2.0 ux) (+ maxCos -1.0)) (pow (* ux (+ maxCos -1.0)) 2.0)))
   (* (cos (* PI (* uy 2.0))) (sqrt (* ux (- 2.0 (* maxCos 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0008549999911338091f) {
		tmp = sqrtf((((-2.0f * ux) * (maxCos + -1.0f)) - powf((ux * (maxCos + -1.0f)), 2.0f)));
	} else {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf((ux * (2.0f - (maxCos * 2.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0008549999911338091))
		tmp = sqrt(Float32(Float32(Float32(Float32(-2.0) * ux) * Float32(maxCos + Float32(-1.0))) - (Float32(ux * Float32(maxCos + Float32(-1.0))) ^ Float32(2.0))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(maxCos * 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(0.0008549999911338091))
		tmp = sqrt((((single(-2.0) * ux) * (maxCos + single(-1.0))) - ((ux * (maxCos + single(-1.0))) ^ single(2.0))));
	else
		tmp = cos((single(pi) * (uy * single(2.0)))) * sqrt((ux * (single(2.0) - (maxCos * single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 8.54999991e-4

   1. Initial program 51.3%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in uy around 0 51.3%

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

      1. associate--l+ 51.1%

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

      2. fma-neg 51.1%

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

      3. mul-1-neg 51.1%

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

      4. fma-def 51.1%

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

      5. distribute-rgt-in 51.1%

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

      6. metadata-eval 51.1%

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

      7. sub-neg 51.1%

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

      8. +-commutative 51.1%

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

      9. sub-neg 51.1%

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

      10. metadata-eval 51.1%

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

      11. *-commutative 51.1%

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

      12. fma-def 51.2%

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

   6. Applied egg-rr 51.2%

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

   7. Taylor expanded in ux around -inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

      3. associate-*r* 98.7%

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

      4. *-commutative 98.7%

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

      5. sub-neg 98.7%

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

      6. metadata-eval 98.7%

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

      7. +-commutative 98.7%

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

      8. unpow2 98.7%

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

      9. unpow2 98.7%

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

      10. swap-sqr 98.7%

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

      11. sub-neg 98.7%

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

      12. metadata-eval 98.7%

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

      13. +-commutative 98.7%

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

      14. sub-neg 98.7%

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

      15. metadata-eval 98.7%

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

      16. +-commutative 98.7%

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

      17. unpow1 98.7%

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

      18. pow-plus 98.7%

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

      19. metadata-eval 98.7%

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

   9. Simplified 98.7%

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

   if 8.54999991e-4 < (*.f32 uy 2)

   1. Initial program 56.9%

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

   2. Taylor expanded in ux around 0 78.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008549999911338091:\\ \;\;\;\;\sqrt{\left(-2 \cdot ux\right) \cdot \left(maxCos + -1\right) - {\left(ux \cdot \left(maxCos + -1\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \end{array} \]

Alternative 10: 80.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.1%

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

3. Simplified 53.2%

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

4. Taylor expanded in uy around 0 45.2%

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

   1. associate--l+ 45.0%

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

   2. fma-neg 45.0%

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

   3. mul-1-neg 45.0%

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

   4. fma-def 45.0%

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

   5. distribute-rgt-in 45.0%

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

   6. metadata-eval 45.0%

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

   7. sub-neg 45.0%

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

   8. +-commutative 45.0%

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

   9. sub-neg 45.0%

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

   10. metadata-eval 45.0%

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

   11. *-commutative 45.0%

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

   12. fma-def 45.1%

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

6. Applied egg-rr 45.1%

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

7. Taylor expanded in ux around -inf 81.9%

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

   1. mul-1-neg 81.9%

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

   2. unsub-neg 81.9%

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

   3. associate-*r* 81.9%

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

   4. *-commutative 81.9%

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

   5. sub-neg 81.9%

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

   6. metadata-eval 81.9%

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

   7. +-commutative 81.9%

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

   8. unpow2 81.9%

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

   9. unpow2 81.9%

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

   10. swap-sqr 81.9%

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

   11. sub-neg 81.9%

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

   12. metadata-eval 81.9%

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

   13. +-commutative 81.9%

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

   14. sub-neg 81.9%

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

   15. metadata-eval 81.9%

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

   16. +-commutative 81.9%

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

   17. unpow1 81.9%

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

   18. pow-plus 81.9%

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

   19. metadata-eval 81.9%

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

9. Simplified 81.9%

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

10. Final simplification 81.9%

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

Alternative 11: 75.8% 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 53.1%

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

3. Simplified 53.2%

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

4. Taylor expanded in ux around 0 99.2%

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

   1. fma-def 99.2%

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

   2. +-commutative 99.2%

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

   3. sub-neg 99.2%

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

   4. metadata-eval 99.2%

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

   5. +-commutative 99.2%

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

   6. distribute-lft-in 99.2%

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

   7. metadata-eval 99.2%

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

   8. associate--l+ 99.2%

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

   9. mul-1-neg 99.2%

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

   10. sub-neg 99.2%

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

   11. *-commutative 99.2%

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

   12. sub-neg 99.2%

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

   13. metadata-eval 99.2%

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

6. Simplified 99.2%

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

7. Taylor expanded in maxCos around 0 94.4%

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

   1. +-commutative 94.4%

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

   2. mul-1-neg 94.4%

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

   3. unsub-neg 94.4%

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

   4. *-commutative 94.4%

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

9. Simplified 94.4%

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

10. Taylor expanded in uy around 0 78.1%

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

    1. *-commutative 78.1%

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

12. Simplified 78.1%

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

13. Final simplification 78.1%

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

Alternative 12: 75.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(1 - maxCos\right)\\ \mathbf{if}\;ux \leq 0.00012700000661425292:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - t_0\right) \cdot \left(-1 + t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 1.0 maxCos))))
   (if (<= ux 0.00012700000661425292)
     (sqrt (* ux (- 2.0 (* maxCos 2.0))))
     (sqrt (+ 1.0 (* (- 1.0 t_0) (+ -1.0 t_0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (1.0f - maxCos);
	float tmp;
	if (ux <= 0.00012700000661425292f) {
		tmp = sqrtf((ux * (2.0f - (maxCos * 2.0f))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - t_0) * (-1.0f + 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 = ux * (1.0e0 - maxcos)
    if (ux <= 0.00012700000661425292e0) then
        tmp = sqrt((ux * (2.0e0 - (maxcos * 2.0e0))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - t_0) * ((-1.0e0) + t_0))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(1.0) - maxCos))
	tmp = Float32(0.0)
	if (ux <= Float32(0.00012700000661425292))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(maxCos * Float32(2.0)))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - t_0) * Float32(Float32(-1.0) + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = ux * (single(1.0) - maxCos);
	tmp = single(0.0);
	if (ux <= single(0.00012700000661425292))
		tmp = sqrt((ux * (single(2.0) - (maxCos * single(2.0)))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - t_0) * (single(-1.0) + t_0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.27000007e-4

   1. Initial program 34.6%

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

   2. Taylor expanded in ux around 0 93.4%

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

   3. Taylor expanded in uy around 0 80.3%

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

   if 1.27000007e-4 < ux

   1. Initial program 89.1%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in uy around 0 72.0%

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

   5. Taylor expanded in ux around -inf 72.0%

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

      1. mul-1-neg 72.0%

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

      2. unsub-neg 72.0%

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

      3. mul-1-neg 72.0%

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

      4. sub-neg 72.0%

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

   7. Simplified 72.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00012700000661425292:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux \cdot \left(1 - maxCos\right)\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]

Alternative 13: 74.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00012700000661425292:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00012700000661425292)
   (sqrt (* ux (- 2.0 (* maxCos 2.0))))
   (sqrt (+ 1.0 (* (- 1.0 ux) (+ -1.0 (* ux (- 1.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00012700000661425292f) {
		tmp = sqrtf((ux * (2.0f - (maxCos * 2.0f))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (-1.0f + (ux * (1.0f - maxCos))))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00012700000661425292e0) then
        tmp = sqrt((ux * (2.0e0 - (maxcos * 2.0e0))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * ((-1.0e0) + (ux * (1.0e0 - maxcos))))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00012700000661425292))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(maxCos * Float32(2.0)))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(Float32(-1.0) + Float32(ux * Float32(Float32(1.0) - maxCos))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00012700000661425292))
		tmp = sqrt((ux * (single(2.0) - (maxCos * single(2.0)))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (single(-1.0) + (ux * (single(1.0) - maxCos))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.27000007e-4

   1. Initial program 34.6%

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

   2. Taylor expanded in ux around 0 93.4%

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

   3. Taylor expanded in uy around 0 80.3%

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

   if 1.27000007e-4 < ux

   1. Initial program 89.1%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in uy around 0 72.0%

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

   5. Taylor expanded in maxCos around 0 69.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00012700000661425292:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]

Alternative 14: 74.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00012700000661425292:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00012700000661425292)
   (sqrt (* ux (- 2.0 (* maxCos 2.0))))
   (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00012700000661425292f) {
		tmp = sqrtf((ux * (2.0f - (maxCos * 2.0f))));
	} 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 <= 0.00012700000661425292e0) then
        tmp = sqrt((ux * (2.0e0 - (maxcos * 2.0e0))))
    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(0.00012700000661425292))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(maxCos * Float32(2.0)))));
	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(0.00012700000661425292))
		tmp = sqrt((ux * (single(2.0) - (maxCos * single(2.0)))));
	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 < 1.27000007e-4

   1. Initial program 34.6%

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

   2. Taylor expanded in ux around 0 93.4%

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

   3. Taylor expanded in uy around 0 80.3%

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

   if 1.27000007e-4 < ux

   1. Initial program 89.1%

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

   3. Simplified 89.3%

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

   4. Taylor expanded in uy around 0 72.0%

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

   5. Taylor expanded in maxCos around 0 69.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00012700000661425292:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \]

Alternative 15: 64.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.1%

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

2. Taylor expanded in ux around 0 79.6%

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

3. Taylor expanded in uy around 0 68.3%

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

4. Final simplification 68.3%

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

Alternative 16: 62.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.1%

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

2. Taylor expanded in ux around 0 79.6%

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

3. Taylor expanded in uy around 0 68.3%

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

4. Taylor expanded in maxCos around 0 66.1%

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

   1. *-commutative 66.1%

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

6. Simplified 66.1%

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

7. Final simplification 66.1%

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

Reproduce

herbie shell --seed 2023292 
(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)))))))