UniformSampleCone, x

Percentage Accurate: 57.2% → 99.0%

Time: 21.2s

Alternatives: 10

Speedup: 3.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot {\left(maxCos + -1\right)}^{2}\right)\right)}^{0.5} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (pow (* ux (- (fma -2.0 maxCos 2.0) (* ux (pow (+ maxCos -1.0) 2.0)))) 0.5)
  (cos (* 2.0 (* uy PI)))))

C

float code(float ux, float uy, float maxCos) {
	return powf((ux * (fmaf(-2.0f, maxCos, 2.0f) - (ux * powf((maxCos + -1.0f), 2.0f)))), 0.5f) * cosf((2.0f * (uy * ((float) M_PI))));
}

Julia

function code(ux, uy, maxCos)
	return Float32((Float32(ux * Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) - Float32(ux * (Float32(maxCos + Float32(-1.0)) ^ Float32(2.0))))) ^ Float32(0.5)) * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))
end

Derivation

1. Initial program 52.7%

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

   1. associate-*l* 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+r- 52.6%

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

   4. fma-def 52.6%

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

   5. +-commutative 52.6%

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

   6. associate-+r- 52.6%

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

   7. fma-def 52.6%

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

3. Simplified 52.6%

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

4. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. +-commutative 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. fma-def 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. unpow2 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in uy around inf 98.9%

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

   1. *-commutative 98.9%

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

9. Simplified 99.0%

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

    1. pow1/2 99.0%

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

    2. *-commutative 99.0%

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

11. Applied egg-rr 99.0%

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

12. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 52.7%

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

   1. associate-*l* 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+r- 52.6%

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

   4. fma-def 52.6%

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

   5. +-commutative 52.6%

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

   6. associate-+r- 52.6%

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

   7. fma-def 52.6%

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

3. Simplified 52.6%

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

4. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. +-commutative 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. fma-def 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. unpow2 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in uy around inf 98.9%

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

   1. *-commutative 98.9%

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

9. Simplified 99.0%

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

    1. pow1/2 99.0%

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

    2. *-commutative 99.0%

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

11. Applied egg-rr 99.0%

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

12. Taylor expanded in uy around inf 99.0%

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

13. Final simplification 99.0%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.7%

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

   1. associate-*l* 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+r- 52.6%

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

   4. fma-def 52.6%

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

   5. +-commutative 52.6%

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

   6. associate-+r- 52.6%

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

   7. fma-def 52.6%

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

3. Simplified 52.6%

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

4. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. +-commutative 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. fma-def 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. unpow2 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in uy around inf 98.9%

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

   1. *-commutative 98.9%

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

9. Simplified 99.0%

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

10. Taylor expanded in maxCos around 0 98.3%

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

11. Final simplification 98.3%

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

Alternative 4: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot {\left(ux \cdot \left(2 - ux\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 3.999999989900971e-6)
   (* (cos (* 2.0 (* uy PI))) (pow (* ux (- 2.0 ux)) 0.5))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux (- 2.0 (* maxCos 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 3.999999989900971e-6f) {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * powf((ux * (2.0f - ux)), 0.5f);
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f - (maxCos * 2.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(3.999999989900971e-6))
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * (Float32(ux * Float32(Float32(2.0) - ux)) ^ Float32(0.5)));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(maxCos * Float32(2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(3.999999989900971e-6))
		tmp = cos((single(2.0) * (uy * single(pi)))) * ((ux * (single(2.0) - ux)) ^ single(0.5));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((ux * (single(2.0) - (maxCos * single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 3.99999999e-6

   1. Initial program 53.7%

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

      1. associate-*l* 53.7%

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

      2. +-commutative 53.7%

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

      3. associate-+r- 53.7%

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

      4. fma-def 53.7%

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

      5. +-commutative 53.7%

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

      6. associate-+r- 53.7%

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

      7. fma-def 53.7%

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

   3. Simplified 53.7%

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

   4. Taylor expanded in ux around 0 98.9%

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

      1. cancel-sign-sub-inv 98.9%

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

      2. metadata-eval 98.9%

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

      3. +-commutative 98.9%

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

      4. mul-1-neg 98.9%

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

      5. unsub-neg 98.9%

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

      6. *-commutative 98.9%

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

      7. fma-def 98.9%

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

      8. sub-neg 98.9%

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

      9. metadata-eval 98.9%

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

      10. unpow2 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in uy around inf 98.9%

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

      1. *-commutative 98.9%

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

   9. Simplified 98.9%

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

       1. pow1/2 98.9%

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

       2. *-commutative 98.9%

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

   11. Applied egg-rr 98.9%

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

   12. Taylor expanded in maxCos around 0 98.7%

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

   if 3.99999999e-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- 46.8%

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

      4. fma-def 46.8%

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

      5. +-commutative 46.8%

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

      6. associate-+r- 46.5%

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

      7. fma-def 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in ux around 0 81.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot {\left(ux \cdot \left(2 - ux\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\\ \end{array} \]

Alternative 5: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot {\left(ux \cdot \left(2 - ux\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (cos (* 2.0 (* uy PI))) (pow (* ux (- 2.0 ux)) 0.5)))

C

float code(float ux, float uy, float maxCos) {
	return cosf((2.0f * (uy * ((float) M_PI)))) * powf((ux * (2.0f - ux)), 0.5f);
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * (Float32(ux * Float32(Float32(2.0) - ux)) ^ Float32(0.5)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = cos((single(2.0) * (uy * single(pi)))) * ((ux * (single(2.0) - ux)) ^ single(0.5));
end

Derivation

1. Initial program 52.7%

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

   1. associate-*l* 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+r- 52.6%

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

   4. fma-def 52.6%

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

   5. +-commutative 52.6%

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

   6. associate-+r- 52.6%

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

   7. fma-def 52.6%

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

3. Simplified 52.6%

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

4. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. +-commutative 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. fma-def 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. unpow2 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in uy around inf 98.9%

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

   1. *-commutative 98.9%

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

9. Simplified 99.0%

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

    1. pow1/2 99.0%

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

    2. *-commutative 99.0%

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

11. Applied egg-rr 99.0%

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

12. Taylor expanded in maxCos around 0 92.5%

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

13. Final simplification 92.5%

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

Alternative 6: 92.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.7%

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

   1. associate-*l* 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+r- 52.6%

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

   4. fma-def 52.6%

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

   5. +-commutative 52.6%

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

   6. associate-+r- 52.6%

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

   7. fma-def 52.6%

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

3. Simplified 52.6%

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

4. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. +-commutative 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. fma-def 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. unpow2 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in maxCos around 0 92.5%

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

   1. unpow2 92.5%

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

   2. distribute-rgt-out-- 92.5%

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

9. Simplified 92.5%

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

10. Final simplification 92.5%

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

Alternative 7: 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 52.7%

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

   1. associate-*l* 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+r- 52.6%

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

   4. fma-def 52.6%

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

   5. +-commutative 52.6%

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

   6. associate-+r- 52.6%

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

   7. fma-def 52.6%

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

3. Simplified 52.6%

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

4. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. +-commutative 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. fma-def 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. unpow2 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in uy around inf 98.9%

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

   1. *-commutative 98.9%

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

9. Simplified 99.0%

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

    1. pow1/2 99.0%

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

    2. *-commutative 99.0%

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

11. Applied egg-rr 99.0%

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

12. Taylor expanded in uy around 0 76.1%

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

13. Final simplification 76.1%

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

Alternative 8: 79.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 52.7%

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

   1. associate-*l* 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+r- 52.6%

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

   4. fma-def 52.6%

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

   5. +-commutative 52.6%

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

   6. associate-+r- 52.6%

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

   7. fma-def 52.6%

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

3. Simplified 52.6%

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

4. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. +-commutative 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. fma-def 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. unpow2 98.9%

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

6. Simplified 98.9%

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

   1. pow1/2 98.9%

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

8. Applied egg-rr 98.9%

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

9. Taylor expanded in uy around 0 76.1%

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

    1. *-commutative 76.1%

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

    2. +-commutative 76.1%

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

    3. *-commutative 76.1%

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

    4. fma-def 76.1%

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

    5. sub-neg 76.1%

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

    6. metadata-eval 76.1%

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

    7. *-commutative 76.1%

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

    8. unpow2 76.1%

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

11. Simplified 76.1%

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

12. Taylor expanded in maxCos around 0 75.2%

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

    1. unpow2 75.2%

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

14. Simplified 75.2%

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

15. Final simplification 75.2%

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

Alternative 9: 75.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 52.7%

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

   1. associate-*l* 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+r- 52.6%

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

   4. fma-def 52.6%

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

   5. +-commutative 52.6%

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

   6. associate-+r- 52.6%

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

   7. fma-def 52.6%

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

3. Simplified 52.6%

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

4. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. +-commutative 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. fma-def 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. unpow2 98.9%

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

6. Simplified 98.9%

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

   1. pow1/2 98.9%

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

8. Applied egg-rr 98.9%

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

9. Taylor expanded in uy around 0 76.1%

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

    1. *-commutative 76.1%

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

    2. +-commutative 76.1%

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

    3. *-commutative 76.1%

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

    4. fma-def 76.1%

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

    5. sub-neg 76.1%

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

    6. metadata-eval 76.1%

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

    7. *-commutative 76.1%

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

    8. unpow2 76.1%

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

11. Simplified 76.1%

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

12. Taylor expanded in maxCos around 0 72.5%

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

    1. unpow2 72.5%

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

    2. distribute-rgt-out-- 72.5%

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

14. Simplified 72.5%

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

15. Final simplification 72.5%

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

Alternative 10: 62.1% 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 52.7%

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

   1. associate-*l* 52.7%

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

   2. +-commutative 52.7%

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

   3. associate-+r- 52.6%

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

   4. fma-def 52.6%

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

   5. +-commutative 52.6%

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

   6. associate-+r- 52.6%

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

   7. fma-def 52.6%

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

3. Simplified 52.6%

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

4. Taylor expanded in uy around 0 44.6%

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

5. Taylor expanded in ux around 0 63.4%

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

6. Taylor expanded in maxCos around 0 61.5%

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

7. Final simplification 61.5%

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

Reproduce

herbie shell --seed 2023279 
(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)))))))