UniformSampleCone, x

Percentage Accurate: 57.9% → 99.0%

Time: 18.7s

Alternatives: 11

Speedup: 2.2×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (* ux (+ 2.0 (- (* -2.0 maxCos) (* ux (pow (+ maxCos -1.0) 2.0))))))
  (cos (* uy (* 2.0 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((uy * (2.0f * ((float) M_PI))));
}

Julia

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

Derivation

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. Add Preprocessing

3. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

   7. metadata-eval 98.9%

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

   8. *-commutative 98.9%

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

5. Simplified 98.9%

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

   1. add-exp-log 98.9%

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

   2. *-commutative 98.9%

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

7. Applied egg-rr 98.9%

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

   1. expm1-log1p-u 98.9%

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

   2. rem-exp-log 98.9%

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

   3. associate-*r* 98.9%

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

9. Applied egg-rr 98.9%

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

10. Taylor expanded in uy around inf 98.9%

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

    1. mul-1-neg 98.9%

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

    2. unsub-neg 98.9%

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

    3. sub-neg 98.9%

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

    4. metadata-eval 98.9%

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

    5. +-commutative 98.9%

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

    6. associate-*r* 98.9%

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

    7. *-commutative 98.9%

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

    8. associate-*l* 98.9%

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

    9. *-commutative 98.9%

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

12. Simplified 98.9%

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

13. Final simplification 98.9%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

3. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

   7. metadata-eval 98.9%

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

   8. *-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in maxCos around 0 98.4%

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

7. Final simplification 98.4%

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

Alternative 3: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0009599999757483602:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(-2 \cdot maxCos + ux \cdot \left(-1 + maxCos \cdot \left(2 - maxCos\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(ux + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0009599999757483602)
   (sqrt
    (*
     ux
     (+ 2.0 (+ (* -2.0 maxCos) (* ux (+ -1.0 (* maxCos (- 2.0 maxCos))))))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux (+ ux 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0009599999757483602f) {
		tmp = sqrtf((ux * (2.0f + ((-2.0f * maxCos) + (ux * (-1.0f + (maxCos * (2.0f - maxCos))))))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (ux + 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0009599999757483602))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(Float32(-2.0) * maxCos) + Float32(ux * Float32(Float32(-1.0) + Float32(maxCos * Float32(Float32(2.0) - maxCos))))))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(ux + Float32(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0009599999757483602))
		tmp = sqrt((ux * (single(2.0) + ((single(-2.0) * maxCos) + (ux * (single(-1.0) + (maxCos * (single(2.0) - maxCos))))))));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((ux * (ux + single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 9.59999976e-4

   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. Add Preprocessing

   3. Taylor expanded in ux around 0 99.4%

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

      1. cancel-sign-sub-inv 99.4%

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

      2. associate-*r* 99.4%

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

      3. mul-1-neg 99.4%

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

      4. sub-neg 99.4%

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

      5. metadata-eval 99.4%

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

      6. +-commutative 99.4%

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

      7. metadata-eval 99.4%

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

      8. *-commutative 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in uy around 0 97.3%

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

   7. Taylor expanded in maxCos around 0 97.3%

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

   if 9.59999976e-4 < uy

   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. Add Preprocessing

   3. Taylor expanded in ux around 0 98.0%

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

      1. cancel-sign-sub-inv 98.0%

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

      2. associate-*r* 98.0%

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

      3. mul-1-neg 98.0%

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

      4. sub-neg 98.0%

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

      5. metadata-eval 98.0%

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

      6. +-commutative 98.0%

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

      7. metadata-eval 98.0%

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

      8. *-commutative 98.0%

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

   5. Simplified 98.0%

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

      1. add-exp-log 98.0%

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

      2. *-commutative 98.0%

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

   7. Applied egg-rr 98.0%

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

   8. Taylor expanded in maxCos around 0 95.2%

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

      1. neg-mul-1 95.2%

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

      2. unsub-neg 95.2%

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

   10. Simplified 95.2%

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

       1. pow1 95.2%

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

       2. *-commutative 95.2%

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

       3. sub-neg 95.2%

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

       4. add-sqr-sqrt -0.0%

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

       5. sqrt-unprod 73.8%

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

       6. sqr-neg 73.8%

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

       7. sqrt-unprod 73.8%

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

       8. add-sqr-sqrt 73.8%

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

       9. rem-exp-log 73.8%

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

   12. Applied egg-rr 73.8%

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

       1. unpow1 73.8%

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

       2. *-commutative 73.8%

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

       3. *-commutative 73.8%

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

       4. associate-*l* 73.8%

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

       5. *-commutative 73.8%

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

   14. Simplified 73.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0009599999757483602:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(-2 \cdot maxCos + ux \cdot \left(-1 + maxCos \cdot \left(2 - maxCos\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(ux + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

3. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

   7. metadata-eval 98.9%

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

   8. *-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in maxCos around 0 94.8%

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

   1. neg-mul-1 94.8%

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

   2. unsub-neg 94.8%

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

8. Simplified 94.8%

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

9. Final simplification 94.8%

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

Alternative 5: 79.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

3. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

   7. metadata-eval 98.9%

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

   8. *-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in uy around 0 81.1%

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

7. Taylor expanded in maxCos around 0 81.1%

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

8. Final simplification 81.1%

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

Alternative 6: 79.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

3. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

   7. metadata-eval 98.9%

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

   8. *-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in uy around 0 81.1%

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

7. Taylor expanded in maxCos around 0 80.8%

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

8. Final simplification 80.8%

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

Alternative 7: 78.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

3. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

   7. metadata-eval 98.9%

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

   8. *-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in uy around 0 81.1%

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

7. Taylor expanded in maxCos around 0 80.4%

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

8. Final simplification 80.4%

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

Alternative 8: 75.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. Add Preprocessing

3. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

   7. metadata-eval 98.9%

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

   8. *-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in uy around 0 81.1%

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

7. Taylor expanded in maxCos around 0 78.1%

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

   1. mul-1-neg 78.1%

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

   2. unsub-neg 78.1%

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

9. Simplified 78.1%

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

10. Final simplification 78.1%

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

Alternative 9: 61.8% accurate, 2.2× 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 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. Add Preprocessing

3. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

   7. metadata-eval 98.9%

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

   8. *-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in uy around 0 81.1%

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

7. Taylor expanded in maxCos around 0 78.1%

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

   1. mul-1-neg 78.1%

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

   2. unsub-neg 78.1%

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

9. Simplified 78.1%

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

10. Taylor expanded in ux around 0 64.9%

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

11. Final simplification 64.9%

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

Alternative 10: 19.4% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(\left(-ux\right) - -0.5 \cdot \frac{2 + ux \cdot 2}{maxCos}\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* maxCos (- (- ux) (* -0.5 (/ (+ 2.0 (* ux 2.0)) maxCos)))))

C

float code(float ux, float uy, float maxCos) {
	return maxCos * (-ux - (-0.5f * ((2.0f + (ux * 2.0f)) / maxCos)));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (-ux - ((-0.5e0) * ((2.0e0 + (ux * 2.0e0)) / maxcos)))
end function

Julia

function code(ux, uy, maxCos)
	return Float32(maxCos * Float32(Float32(-ux) - Float32(Float32(-0.5) * Float32(Float32(Float32(2.0) + Float32(ux * Float32(2.0))) / maxCos))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = maxCos * (-ux - (single(-0.5) * ((single(2.0) + (ux * single(2.0))) / maxCos)));
end

Derivation

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. Add Preprocessing

3. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

   7. metadata-eval 98.9%

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

   8. *-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in uy around 0 81.1%

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

   1. *-commutative 81.1%

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

   2. +-commutative 81.1%

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

   3. associate-*r* 81.1%

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

   4. neg-mul-1 81.1%

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

   5. sub-neg 81.1%

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

   6. metadata-eval 81.1%

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

   7. +-commutative 81.1%

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

   8. *-commutative 81.1%

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

   9. associate-+l+ 81.1%

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

   10. *-commutative 81.1%

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

   11. add-cube-cbrt 80.0%

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

8. Applied egg-rr 64.5%

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

9. Taylor expanded in maxCos around -inf 19.5%

   \[\leadsto \color{blue}{-1 \cdot \left(maxCos \cdot \left(ux + -0.5 \cdot \frac{2 + 2 \cdot ux}{maxCos}\right)\right)} \]
10. Step-by-step derivation

    1. mul-1-neg 19.5%

       \[\leadsto \color{blue}{-maxCos \cdot \left(ux + -0.5 \cdot \frac{2 + 2 \cdot ux}{maxCos}\right)} \]

    2. *-commutative 19.5%

       \[\leadsto -\color{blue}{\left(ux + -0.5 \cdot \frac{2 + 2 \cdot ux}{maxCos}\right) \cdot maxCos} \]

    3. distribute-rgt-neg-in 19.5%

       \[\leadsto \color{blue}{\left(ux + -0.5 \cdot \frac{2 + 2 \cdot ux}{maxCos}\right) \cdot \left(-maxCos\right)} \]

11. Simplified 19.5%

    \[\leadsto \color{blue}{\left(ux + -0.5 \cdot \frac{2 + 2 \cdot ux}{maxCos}\right) \cdot \left(-maxCos\right)} \]

12. Final simplification 19.5%

    \[\leadsto maxCos \cdot \left(\left(-ux\right) - -0.5 \cdot \frac{2 + ux \cdot 2}{maxCos}\right) \]
13. Add Preprocessing

Alternative 11: 9.6% accurate, 74.3× speedup

Math

\[\begin{array}{l} \\ ux \cdot maxCos \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return ux * 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 = ux * maxcos
end function

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

3. Taylor expanded in ux around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. associate-*r* 98.9%

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

   3. mul-1-neg 98.9%

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

   4. sub-neg 98.9%

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

   5. metadata-eval 98.9%

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

   6. +-commutative 98.9%

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

   7. metadata-eval 98.9%

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

   8. *-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in uy around 0 81.1%

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

   1. *-commutative 81.1%

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

   2. +-commutative 81.1%

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

   3. associate-*r* 81.1%

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

   4. neg-mul-1 81.1%

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

   5. sub-neg 81.1%

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

   6. metadata-eval 81.1%

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

   7. +-commutative 81.1%

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

   8. *-commutative 81.1%

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

   9. associate-+l+ 81.1%

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

   10. *-commutative 81.1%

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

   11. add-cube-cbrt 80.0%

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

8. Applied egg-rr 64.5%

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

9. Taylor expanded in maxCos around inf 9.4%

   \[\leadsto \color{blue}{maxCos \cdot ux} \]
10. Step-by-step derivation

    1. *-commutative 9.4%

       \[\leadsto \color{blue}{ux \cdot maxCos} \]

11. Simplified 9.4%

    \[\leadsto \color{blue}{ux \cdot maxCos} \]

12. Final simplification 9.4%

    \[\leadsto ux \cdot maxCos \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(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)))))))