UniformSampleCone, x

Average Error: 13.9 → 0.3

Time: 21.1s

Precision: binary32

Cost: 13408

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Derivation

1. Initial program 13.9

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

2. Simplified 13.9

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

   [Start] 13.9	\[ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   associate-*l* [=>] 13.9	\[ \cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   cancel-sign-sub-inv [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{1 + \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]
   +-commutative [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + 1}} \]
   *-commutative [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]
   fma-def [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 - ux\right) + ux \cdot maxCos, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)}} \]
   +-commutative [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{ux \cdot maxCos + \left(1 - ux\right)}, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   associate-+r- [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\left(ux \cdot maxCos + 1\right) - ux}, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   fma-def [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)} - ux, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   neg-sub0 [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]
   +-commutative [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]
   associate-+r- [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]
   associate--r- [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]
   +-commutative [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux + \left(0 - \left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]
   sub0-neg [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux + \color{blue}{\left(-\left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]
   sub-neg [<=] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux - \left(ux \cdot maxCos + 1\right)}, 1\right)} \]
   fma-def [=>] 13.9	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux - \color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)}, 1\right)} \]

3. Taylor expanded in ux around -inf 0.3

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

4. Simplified 0.3

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

   [Start] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{-1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right) + 2 \cdot \left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right)} \]
   +-commutative [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right) + -1 \cdot \left({ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)}} \]
   mul-1-neg [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot \left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)}} \]
   unsub-neg [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right) - {ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}}} \]
   *-commutative [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right) \cdot 2} - {ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}} \]
   mul-1-neg [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux \cdot \left(1 + \color{blue}{\left(-maxCos\right)}\right)\right) \cdot 2 - {ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}} \]
   sub-neg [<=] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux \cdot \color{blue}{\left(1 - maxCos\right)}\right) \cdot 2 - {ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}} \]
   associate-*l* [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(1 - maxCos\right) \cdot 2\right)} - {ux}^{2} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}} \]
   unpow2 [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot 2\right) - \color{blue}{\left(ux \cdot ux\right)} \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}} \]
   associate-*l* [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot 2\right) - \color{blue}{ux \cdot \left(ux \cdot {\left(1 + -1 \cdot maxCos\right)}^{2}\right)}} \]
   mul-1-neg [=>] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot 2\right) - ux \cdot \left(ux \cdot {\left(1 + \color{blue}{\left(-maxCos\right)}\right)}^{2}\right)} \]
   sub-neg [<=] 0.3	\[ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) \cdot 2\right) - ux \cdot \left(ux \cdot {\color{blue}{\left(1 - maxCos\right)}}^{2}\right)} \]

5. Applied egg-rr 0.3

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

6. Taylor expanded in uy around inf 0.3

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

7. Simplified 0.3

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

   [Start] 0.3	\[ \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot \left(\left(1 - maxCos\right) \cdot ux\right) - {\left(1 - maxCos\right)}^{2} \cdot {ux}^{2}} \]
   *-commutative [=>] 0.3	\[ \color{blue}{\sqrt{2 \cdot \left(\left(1 - maxCos\right) \cdot ux\right) - {\left(1 - maxCos\right)}^{2} \cdot {ux}^{2}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
   *-commutative [=>] 0.3	\[ \sqrt{\color{blue}{\left(\left(1 - maxCos\right) \cdot ux\right) \cdot 2} - {\left(1 - maxCos\right)}^{2} \cdot {ux}^{2}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
   associate-*l* [=>] 0.3	\[ \sqrt{\color{blue}{\left(1 - maxCos\right) \cdot \left(ux \cdot 2\right)} - {\left(1 - maxCos\right)}^{2} \cdot {ux}^{2}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
   *-commutative [<=] 0.3	\[ \sqrt{\left(1 - maxCos\right) \cdot \color{blue}{\left(2 \cdot ux\right)} - {\left(1 - maxCos\right)}^{2} \cdot {ux}^{2}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
   *-commutative [=>] 0.3	\[ \sqrt{\left(1 - maxCos\right) \cdot \left(2 \cdot ux\right) - \color{blue}{{ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
   unpow2 [=>] 0.3	\[ \sqrt{\left(1 - maxCos\right) \cdot \left(2 \cdot ux\right) - {ux}^{2} \cdot \color{blue}{\left(\left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
   unpow2 [=>] 0.3	\[ \sqrt{\left(1 - maxCos\right) \cdot \left(2 \cdot ux\right) - \color{blue}{\left(ux \cdot ux\right)} \cdot \left(\left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
   swap-sqr [<=] 0.3	\[ \sqrt{\left(1 - maxCos\right) \cdot \left(2 \cdot ux\right) - \color{blue}{\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
   unpow2 [<=] 0.3	\[ \sqrt{\left(1 - maxCos\right) \cdot \left(2 \cdot ux\right) - \color{blue}{{\left(ux \cdot \left(1 - maxCos\right)\right)}^{2}}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
   *-commutative [<=] 0.3	\[ \sqrt{\left(1 - maxCos\right) \cdot \left(2 \cdot ux\right) - {\color{blue}{\left(\left(1 - maxCos\right) \cdot ux\right)}}^{2}} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
   associate-*r* [=>] 0.3	\[ \sqrt{\left(1 - maxCos\right) \cdot \left(2 \cdot ux\right) - {\left(\left(1 - maxCos\right) \cdot ux\right)}^{2}} \cdot \cos \color{blue}{\left(\left(2 \cdot uy\right) \cdot \pi\right)} \]
   *-commutative [=>] 0.3	\[ \sqrt{\left(1 - maxCos\right) \cdot \left(2 \cdot ux\right) - {\left(\left(1 - maxCos\right) \cdot ux\right)}^{2}} \cdot \cos \color{blue}{\left(\pi \cdot \left(2 \cdot uy\right)\right)} \]

8. Final simplification 0.3

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

Alternatives

Alternative 1
Error	3.4
Cost	16420

\[\begin{array}{l} t_0 := \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999200105667114:\\ \;\;\;\;t_0 \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot ux - maxCos \cdot \left(maxCos \cdot \left(ux \cdot ux\right)\right)\right) - maxCos \cdot \mathsf{fma}\left(-2, ux \cdot ux, 2 \cdot ux\right)\right) - ux \cdot ux}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	10176

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

Alternative 3
Error	0.3
Cost	10176

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

Alternative 4
Error	1.4
Cost	9988

\[\begin{array}{l} \mathbf{if}\;maxCos \leq 3.000000106112566 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot ux - maxCos \cdot \left(maxCos \cdot \left(ux \cdot ux\right)\right)\right) - maxCos \cdot \mathsf{fma}\left(-2, ux \cdot ux, 2 \cdot ux\right)\right) - ux \cdot ux}\\ \end{array} \]

Alternative 5
Error	6.4
Cost	7200

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

Alternative 6
Error	6.4
Cost	6848

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

Alternative 7
Error	6.4
Cost	3616

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

Alternative 8
Error	6.4
Cost	3616

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

Alternative 9
Error	11.2
Cost	3424

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

Alternative 10
Error	7.8
Cost	3424

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

Alternative 11
Error	12.0
Cost	3296

\[\sqrt{ux + ux} \]

Alternative 12
Error	29.9
Cost	3232

\[\sqrt{0} \]

Reproduce

herbie shell --seed 2022354 
(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)))))))