UniformSampleCone, x

Percentage Accurate: 58.4% → 99.0%

Time: 22.9s

Alternatives: 12

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: 58.4% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

   1. associate-*l* 57.3%

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

   2. +-commutative 57.3%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.3%

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

   7. fma-def 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in ux around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. cancel-sign-sub-inv 98.8%

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

   3. metadata-eval 98.8%

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

   4. mul-1-neg 98.8%

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

   5. unsub-neg 98.8%

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

   6. +-commutative 98.8%

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

   7. *-commutative 98.8%

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

   8. fma-def 98.8%

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

   9. unpow2 98.8%

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

   10. associate-*l* 98.8%

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

   11. sub-neg 98.8%

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

   12. metadata-eval 98.8%

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

6. Simplified 98.8%

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

   1. expm1-log1p-u 98.6%

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

   2. distribute-lft-out-- 98.7%

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

8. Applied egg-rr 98.7%

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

9. Taylor expanded in uy around inf 98.9%

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

10. Final simplification 98.9%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

   1. associate-*l* 57.3%

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

   2. +-commutative 57.3%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.3%

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

   7. fma-def 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in ux around -inf 59.7%

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

   1. +-commutative 59.7%

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

   2. mul-1-neg 59.7%

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

   3. unsub-neg 59.7%

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

   4. unpow2 59.7%

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

   5. mul-1-neg 59.7%

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

   6. unsub-neg 59.7%

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

   7. +-commutative 59.7%

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

   8. *-commutative 59.7%

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

   9. fma-def 59.7%

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

6. Simplified 59.7%

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

7. Taylor expanded in maxCos around -inf 98.8%

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

   1. associate-+r+ 98.8%

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

   2. associate--l+ 98.8%

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

9. Simplified 98.8%

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

10. Final simplification 98.8%

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

Alternative 3: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

   1. associate-*l* 57.3%

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

   2. +-commutative 57.3%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.3%

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

   7. fma-def 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in ux around -inf 59.7%

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

   1. +-commutative 59.7%

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

   2. mul-1-neg 59.7%

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

   3. unsub-neg 59.7%

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

   4. unpow2 59.7%

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

   5. mul-1-neg 59.7%

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

   6. unsub-neg 59.7%

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

   7. +-commutative 59.7%

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

   8. *-commutative 59.7%

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

   9. fma-def 59.7%

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

6. Simplified 59.7%

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

7. Taylor expanded in maxCos around 0 98.0%

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

   1. +-commutative 98.0%

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

   2. associate--l+ 98.0%

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

   3. distribute-lft-out-- 98.0%

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

   4. unpow2 98.0%

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

   5. unpow2 98.0%

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

   6. distribute-rgt-out-- 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 4: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

   1. associate-*l* 57.3%

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

   2. +-commutative 57.3%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.3%

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

   7. fma-def 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in ux around -inf 59.7%

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

   1. +-commutative 59.7%

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

   2. mul-1-neg 59.7%

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

   3. unsub-neg 59.7%

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

   4. unpow2 59.7%

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

   5. mul-1-neg 59.7%

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

   6. unsub-neg 59.7%

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

   7. +-commutative 59.7%

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

   8. *-commutative 59.7%

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

   9. fma-def 59.7%

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

6. Simplified 59.7%

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

7. Taylor expanded in maxCos around -inf 98.8%

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

   1. associate-+r+ 98.8%

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

   2. associate--l+ 98.8%

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

9. Simplified 98.8%

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

10. Taylor expanded in ux around 0 97.5%

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

    1. *-commutative 97.5%

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

12. Simplified 97.5%

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

13. Final simplification 97.5%

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

Alternative 5: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 + -2 \cdot maxCos\right) - ux \cdot {\left(maxCos - 1\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 7.999999979801942e-6)
   (* (cos (* PI (* 2.0 uy))) (sqrt (* ux (- 2.0 ux))))
   (sqrt (* ux (- (+ 2.0 (* -2.0 maxCos)) (* ux (pow (- maxCos 1.0) 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 7.999999979801942e-6f) {
		tmp = cosf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf((ux * ((2.0f + (-2.0f * maxCos)) - (ux * powf((maxCos - 1.0f), 2.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(7.999999979801942e-6))
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(Float32(-2.0) * maxCos)) - Float32(ux * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(7.999999979801942e-6))
		tmp = cos((single(pi) * (single(2.0) * uy))) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = sqrt((ux * ((single(2.0) + (single(-2.0) * maxCos)) - (ux * ((maxCos - single(1.0)) ^ single(2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 7.99999998e-6

   1. Initial program 58.7%

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

      1. associate-*l* 58.7%

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

      2. +-commutative 58.7%

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

      3. associate-+r- 58.7%

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

      4. fma-def 58.7%

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

      5. +-commutative 58.7%

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

      6. associate-+r- 58.7%

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

      7. fma-def 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in ux around 0 98.8%

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

      1. +-commutative 98.8%

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

      2. cancel-sign-sub-inv 98.8%

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

      3. metadata-eval 98.8%

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

      4. mul-1-neg 98.8%

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

      5. unsub-neg 98.8%

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

      6. +-commutative 98.8%

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

      7. *-commutative 98.8%

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

      8. fma-def 98.8%

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

      9. unpow2 98.8%

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

      10. associate-*l* 98.8%

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

      11. sub-neg 98.8%

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

      12. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in maxCos around 0 98.0%

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

      1. associate-*r* 98.0%

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

      2. unpow2 98.0%

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

      3. distribute-rgt-out-- 98.1%

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

   9. Simplified 98.1%

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

   if 7.99999998e-6 < maxCos

   1. Initial program 47.3%

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

      1. associate-*l* 47.3%

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

      2. +-commutative 47.3%

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

      3. associate-+r- 47.3%

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

      4. fma-def 47.3%

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

      5. +-commutative 47.3%

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

      6. associate-+r- 47.4%

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

      7. fma-def 47.4%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in ux around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. cancel-sign-sub-inv 99.0%

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

      3. metadata-eval 99.0%

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

      4. mul-1-neg 99.0%

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

      5. unsub-neg 99.0%

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

      6. +-commutative 99.0%

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

      7. *-commutative 99.0%

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

      8. fma-def 99.0%

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

      9. unpow2 99.0%

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

      10. associate-*l* 99.1%

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

      11. sub-neg 99.1%

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

      12. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

      1. expm1-log1p-u 98.9%

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

      2. distribute-lft-out-- 98.9%

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

   8. Applied egg-rr 98.9%

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

   9. Taylor expanded in uy around 0 85.2%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 + -2 \cdot maxCos\right) - ux \cdot {\left(maxCos - 1\right)}^{2}\right)}\\ \end{array} \]

Alternative 6: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0024500000290572643:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right) - maxCos \cdot \mathsf{fma}\left(2, ux - ux \cdot ux, ux \cdot \left(ux \cdot maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0024500000290572643)
   (sqrt
    (-
     (* ux (- 2.0 ux))
     (* maxCos (fma 2.0 (- ux (* ux ux)) (* ux (* ux maxCos))))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0024500000290572643f) {
		tmp = sqrtf(((ux * (2.0f - ux)) - (maxCos * fmaf(2.0f, (ux - (ux * ux)), (ux * (ux * maxCos))))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0024500000290572643))
		tmp = sqrt(Float32(Float32(ux * Float32(Float32(2.0) - ux)) - Float32(maxCos * fma(Float32(2.0), Float32(ux - Float32(ux * ux)), Float32(ux * Float32(ux * maxCos))))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00245000003

   1. Initial program 59.4%

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

      1. associate-*l* 59.4%

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

      2. +-commutative 59.4%

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

      3. associate-+r- 59.5%

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

      4. fma-def 59.5%

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

      5. +-commutative 59.5%

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

      6. associate-+r- 59.5%

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

      7. fma-def 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in ux around -inf 61.8%

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

      1. +-commutative 61.8%

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

      2. mul-1-neg 61.8%

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

      3. unsub-neg 61.8%

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

      4. unpow2 61.8%

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

      5. mul-1-neg 61.8%

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

      6. unsub-neg 61.8%

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

      7. +-commutative 61.8%

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

      8. *-commutative 61.8%

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

      9. fma-def 61.8%

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

   6. Simplified 61.8%

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

   7. Taylor expanded in maxCos around -inf 99.3%

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

      1. associate-+r+ 99.3%

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

      2. associate--l+ 99.3%

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

   9. Simplified 99.3%

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

   10. Taylor expanded in uy around 0 95.6%

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

       1. cancel-sign-sub-inv 95.6%

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

       2. fma-def 95.6%

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

       3. unpow2 95.6%

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

       4. *-commutative 95.6%

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

       5. unpow2 95.6%

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

       6. associate-*r* 95.6%

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

       7. cancel-sign-sub-inv 95.6%

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

   12. Simplified 95.6%

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

   if 0.00245000003 < uy

   1. Initial program 51.5%

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

      1. associate-*l* 51.5%

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

      2. +-commutative 51.5%

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

      3. associate-+r- 51.5%

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

      4. fma-def 51.5%

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

      5. +-commutative 51.5%

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

      6. associate-+r- 51.4%

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

      7. fma-def 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in ux around 0 41.2%

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

   5. Taylor expanded in maxCos around 0 76.9%

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

      1. *-commutative 76.9%

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

   7. Simplified 76.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0024500000290572643:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right) - maxCos \cdot \mathsf{fma}\left(2, ux - ux \cdot ux, ux \cdot \left(ux \cdot maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]

Alternative 7: 80.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

   1. associate-*l* 57.3%

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

   2. +-commutative 57.3%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.3%

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

   7. fma-def 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in ux around -inf 59.7%

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

   1. +-commutative 59.7%

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

   2. mul-1-neg 59.7%

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

   3. unsub-neg 59.7%

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

   4. unpow2 59.7%

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

   5. mul-1-neg 59.7%

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

   6. unsub-neg 59.7%

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

   7. +-commutative 59.7%

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

   8. *-commutative 59.7%

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

   9. fma-def 59.7%

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

6. Simplified 59.7%

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

7. Taylor expanded in maxCos around -inf 98.8%

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

   1. associate-+r+ 98.8%

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

   2. associate--l+ 98.8%

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

9. Simplified 98.8%

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

10. Taylor expanded in uy around 0 80.0%

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

    1. cancel-sign-sub-inv 80.0%

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

    2. fma-def 80.0%

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

    3. unpow2 80.0%

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

    4. *-commutative 80.0%

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

    5. unpow2 80.0%

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

    6. associate-*r* 80.0%

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

    7. cancel-sign-sub-inv 80.0%

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

12. Simplified 80.0%

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

13. Final simplification 80.0%

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

Alternative 8: 80.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

   1. associate-*l* 57.3%

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

   2. +-commutative 57.3%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.3%

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

   7. fma-def 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in ux around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. cancel-sign-sub-inv 98.8%

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

   3. metadata-eval 98.8%

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

   4. mul-1-neg 98.8%

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

   5. unsub-neg 98.8%

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

   6. +-commutative 98.8%

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

   7. *-commutative 98.8%

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

   8. fma-def 98.8%

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

   9. unpow2 98.8%

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

   10. associate-*l* 98.8%

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

   11. sub-neg 98.8%

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

   12. metadata-eval 98.8%

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

6. Simplified 98.8%

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

   1. expm1-log1p-u 98.6%

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

   2. distribute-lft-out-- 98.7%

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

8. Applied egg-rr 98.7%

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

9. Taylor expanded in uy around 0 80.0%

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

10. Final simplification 80.0%

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

Alternative 9: 75.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 7.999999797903001 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 - -2 \cdot \left(maxCos \cdot \left(ux \cdot \left(ux - 1\right)\right)\right)\right) + \left(-1 - ux \cdot \left(ux + -2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 7.999999797903001e-5)
   (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
   (sqrt
    (+
     (- 1.0 (* -2.0 (* maxCos (* ux (- ux 1.0)))))
     (- -1.0 (* ux (+ ux -2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 7.999999797903001e-5f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf(((1.0f - (-2.0f * (maxCos * (ux * (ux - 1.0f))))) + (-1.0f - (ux * (ux + -2.0f)))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 7.999999797903001e-5) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt(((1.0e0 - ((-2.0e0) * (maxcos * (ux * (ux - 1.0e0))))) + ((-1.0e0) - (ux * (ux + (-2.0e0))))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(7.999999797903001e-5))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(Float32(1.0) - Float32(Float32(-2.0) * Float32(maxCos * Float32(ux * Float32(ux - Float32(1.0)))))) + Float32(Float32(-1.0) - Float32(ux * Float32(ux + Float32(-2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(7.999999797903001e-5))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt(((single(1.0) - (single(-2.0) * (maxCos * (ux * (ux - single(1.0)))))) + (single(-1.0) - (ux * (ux + single(-2.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 7.9999998e-5

   1. Initial program 32.2%

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

      1. associate-*l* 32.2%

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

      2. +-commutative 32.2%

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

      3. associate-+r- 32.2%

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

      4. fma-def 32.2%

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

      5. +-commutative 32.2%

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

      6. associate-+r- 32.2%

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

      7. fma-def 32.2%

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

   3. Simplified 32.2%

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

   4. Taylor expanded in uy around 0 29.5%

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

   5. Taylor expanded in ux around 0 74.8%

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

   if 7.9999998e-5 < ux

   1. Initial program 88.1%

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

      1. associate-*l* 88.1%

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

      2. +-commutative 88.1%

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

      3. associate-+r- 88.2%

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

      4. fma-def 88.2%

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

      5. +-commutative 88.2%

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

      6. associate-+r- 88.2%

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

      7. fma-def 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in uy around 0 75.5%

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

   5. Taylor expanded in maxCos around 0 74.4%

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

   6. Taylor expanded in ux around 0 75.7%

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

      1. +-commutative 75.7%

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

      2. unpow2 75.7%

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

      3. distribute-rgt-out 75.8%

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

   8. Simplified 75.8%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 7.999999797903001 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 - -2 \cdot \left(maxCos \cdot \left(ux \cdot \left(ux - 1\right)\right)\right)\right) + \left(-1 - ux \cdot \left(ux + -2\right)\right)}\\ \end{array} \]

Alternative 10: 74.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if ux < 9.00000014e-5

   1. Initial program 32.4%

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

      1. associate-*l* 32.4%

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

      2. +-commutative 32.4%

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

      3. associate-+r- 32.4%

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

      4. fma-def 32.4%

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

      5. +-commutative 32.4%

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

      6. associate-+r- 32.4%

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

      7. fma-def 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in uy around 0 29.7%

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

   5. Taylor expanded in ux around 0 74.8%

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

   if 9.00000014e-5 < ux

   1. Initial program 88.3%

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

      1. associate-*l* 88.3%

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

      2. +-commutative 88.3%

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

      3. associate-+r- 88.4%

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

      4. fma-def 88.4%

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

      5. +-commutative 88.4%

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

      6. associate-+r- 88.4%

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

      7. fma-def 88.4%

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

   3. Simplified 88.4%

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

      1. pow2 88.4%

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

      2. fma-udef 88.4%

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

      3. associate-+r- 88.3%

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

      4. +-commutative 88.3%

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

      5. pow2 88.3%

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

      6. distribute-lft-in 88.2%

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

      7. +-commutative 88.2%

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

      8. associate-+r- 88.4%

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

      9. fma-udef 88.4%

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

      10. +-commutative 88.4%

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

      11. associate-+r- 88.4%

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

      12. fma-udef 88.4%

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

   5. Applied egg-rr 88.4%

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

      1. distribute-lft-out 88.4%

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

   7. Simplified 88.4%

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

   8. Taylor expanded in uy around 0 75.6%

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

   9. Taylor expanded in maxCos around 0 72.7%

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

4. Final simplification 73.9%

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

Alternative 11: 63.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

   1. associate-*l* 57.3%

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

   2. +-commutative 57.3%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.3%

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

   7. fma-def 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in uy around 0 50.1%

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

5. Taylor expanded in ux around 0 63.4%

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

6. Final simplification 63.4%

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

Alternative 12: 61.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

   1. associate-*l* 57.3%

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

   2. +-commutative 57.3%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.3%

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

   7. fma-def 57.3%

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

3. Simplified 57.3%

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

4. Taylor expanded in uy around 0 50.1%

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

5. Taylor expanded in ux around 0 63.4%

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

6. Taylor expanded in maxCos around 0 60.1%

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

7. Final simplification 60.1%

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

Reproduce

herbie shell --seed 2023278 
(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)))))))