UniformSampleCone, x

Percentage Accurate: 57.9% → 99.0%

Time: 13.2s

Alternatives: 8

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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0 99.0%

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

   1. associate--l+ 99.0%

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

   2. associate-*r* 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in maxCos around 0 99.0%

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

   1. add-cube-cbrt 98.9%

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

   2. pow3 98.9%

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

8. Applied egg-rr 98.9%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}}\right) \cdot \sqrt{ux \cdot \left(2 + \left(-1 \cdot ux + maxCos \cdot \left(\left(-1 \cdot \left(maxCos \cdot ux\right) + 2 \cdot ux\right) - 2\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 98.9%

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

   2. rem-cube-cbrt 99.0%

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

   3. associate-*r* 99.0%

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

   4. cos-2 98.8%

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

   5. prod-diff 98.8%

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

   6. fmm-def 98.8%

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

   7. cos-2 99.0%

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

   8. associate-*r* 99.0%

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

   9. *-commutative 99.0%

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

   10. associate-*l* 99.0%

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

   11. pow2 99.0%

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

10. Applied egg-rr 99.0%

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

11. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0 99.0%

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

   1. associate--l+ 99.0%

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

   2. associate-*r* 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in maxCos around 0 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. neg-mul-1 99.0%

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

   3. +-commutative 99.0%

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

   4. fma-define 99.0%

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

   5. associate--l+ 99.0%

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

   6. fma-define 99.0%

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

   7. *-commutative 99.0%

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

   8. fmm-def 99.0%

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

   9. metadata-eval 99.0%

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

8. Applied egg-rr 99.0%

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

   1. *-lft-identity 99.0%

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

   2. fmm-undef 99.0%

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

   3. fma-undefine 99.0%

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

   4. neg-mul-1 99.0%

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

   5. fma-undefine 99.0%

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

   6. associate-+r+ 99.0%

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

   7. distribute-lft-neg-in 99.0%

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

   8. mul-1-neg 99.0%

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

   9. *-commutative 99.0%

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

   10. distribute-rgt-in 99.0%

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

   11. +-commutative 99.0%

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

   12. mul-1-neg 99.0%

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

   13. unsub-neg 99.0%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 3: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0 99.0%

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

   1. associate--l+ 99.0%

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

   2. associate-*r* 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in maxCos around 0 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. neg-mul-1 99.0%

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

   3. +-commutative 99.0%

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

   4. fma-define 99.0%

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

   5. associate--l+ 99.0%

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

   6. fma-define 99.0%

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

   7. *-commutative 99.0%

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

   8. fmm-def 99.0%

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

   9. metadata-eval 99.0%

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

8. Applied egg-rr 99.0%

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

   1. *-lft-identity 99.0%

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

   2. fmm-undef 99.0%

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

   3. fma-undefine 99.0%

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

   4. neg-mul-1 99.0%

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

   5. fma-undefine 99.0%

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

   6. associate-+r+ 99.0%

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

   7. distribute-lft-neg-in 99.0%

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

   8. mul-1-neg 99.0%

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

   9. *-commutative 99.0%

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

   10. distribute-rgt-in 99.0%

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

   11. +-commutative 99.0%

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

   12. mul-1-neg 99.0%

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

   13. unsub-neg 99.0%

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

10. Simplified 99.0%

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

11. Taylor expanded in maxCos around 0 98.6%

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

12. Final simplification 98.6%

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

Alternative 4: 92.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0 99.0%

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

   1. associate--l+ 99.0%

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

   2. associate-*r* 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in maxCos around 0 94.4%

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

   1. neg-mul-1 94.4%

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

   2. unsub-neg 94.4%

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

8. Simplified 94.4%

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

9. Final simplification 94.4%

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

Alternative 5: 80.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in ux around 0 99.0%

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

   1. associate--l+ 99.0%

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

   2. associate-*r* 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in maxCos around 0 99.0%

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

   1. *-un-lft-identity 99.0%

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

   2. neg-mul-1 99.0%

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

   3. +-commutative 99.0%

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

   4. fma-define 99.0%

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

   5. associate--l+ 99.0%

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

   6. fma-define 99.0%

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

   7. *-commutative 99.0%

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

   8. fmm-def 99.0%

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

   9. metadata-eval 99.0%

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

8. Applied egg-rr 99.0%

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

   1. *-lft-identity 99.0%

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

   2. fmm-undef 99.0%

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

   3. fma-undefine 99.0%

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

   4. neg-mul-1 99.0%

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

   5. fma-undefine 99.0%

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

   6. associate-+r+ 99.0%

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

   7. distribute-lft-neg-in 99.0%

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

   8. mul-1-neg 99.0%

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

   9. *-commutative 99.0%

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

   10. distribute-rgt-in 99.0%

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

   11. +-commutative 99.0%

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

   12. mul-1-neg 99.0%

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

   13. unsub-neg 99.0%

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

10. Simplified 99.0%

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

11. Taylor expanded in uy around 0 79.3%

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

Alternative 6: 75.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ ux \cdot \sqrt{-1 + \frac{2}{ux}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

   1. associate-*l* 57.8%

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

   2. sub-neg 57.8%

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

   3. +-commutative 57.8%

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

   4. distribute-rgt-neg-in 57.8%

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

   5. fma-define 57.8%

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

3. Simplified 57.9%

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

5. Taylor expanded in ux around inf 98.7%

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

6. Taylor expanded in maxCos around 0 93.9%

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

   1. sub-neg 93.9%

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

   2. associate-*r/ 93.9%

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

   3. metadata-eval 93.9%

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

   4. metadata-eval 93.9%

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

8. Simplified 93.9%

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

9. Taylor expanded in uy around 0 76.0%

   \[\leadsto \color{blue}{ux} \cdot \sqrt{\frac{2}{ux} + -1} \]

10. Final simplification 76.0%

    \[\leadsto ux \cdot \sqrt{-1 + \frac{2}{ux}} \]
11. Add Preprocessing

Alternative 7: 64.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

   1. associate-*l* 57.8%

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

   2. sub-neg 57.8%

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

   3. +-commutative 57.8%

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

   4. distribute-rgt-neg-in 57.8%

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

   5. fma-define 57.8%

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

3. Simplified 57.9%

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

5. Taylor expanded in uy around 0 50.0%

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

   1. mul-1-neg 50.0%

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

   2. unsub-neg 50.0%

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

   3. sub-neg 50.0%

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

   4. metadata-eval 50.0%

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

   5. distribute-lft-in 50.0%

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

   6. *-commutative 50.0%

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

   7. mul-1-neg 50.0%

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

   8. sub-neg 50.0%

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

   9. *-commutative 50.0%

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

   10. associate--l+ 49.9%

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

   11. unpow2 49.9%

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

   12. sub-neg 49.9%

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

7. Simplified 50.0%

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

8. Taylor expanded in ux around 0 63.2%

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

Alternative 8: 61.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

   1. associate-*l* 57.8%

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

   2. sub-neg 57.8%

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

   3. +-commutative 57.8%

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

   4. distribute-rgt-neg-in 57.8%

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

   5. fma-define 57.8%

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

3. Simplified 57.9%

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

5. Taylor expanded in uy around 0 50.0%

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

   1. mul-1-neg 50.0%

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

   2. unsub-neg 50.0%

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

   3. sub-neg 50.0%

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

   4. metadata-eval 50.0%

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

   5. distribute-lft-in 50.0%

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

   6. *-commutative 50.0%

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

   7. mul-1-neg 50.0%

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

   8. sub-neg 50.0%

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

   9. *-commutative 50.0%

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

   10. associate--l+ 49.9%

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

   11. unpow2 49.9%

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

   12. sub-neg 49.9%

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

7. Simplified 50.0%

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

8. Taylor expanded in ux around 0 63.2%

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

9. Taylor expanded in maxCos around 0 61.6%

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

10. Final simplification 61.6%

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

Reproduce

herbie shell --seed 2024157 
(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)))))))