UniformSampleCone, y

Average Error: 13.5 → 0.6

Time: 21.4s

Precision: binary32

Cost: 29568

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

Math

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

↓

\[\sin \left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(8 \cdot {\pi}^{3}\right)\right) \cdot {uy}^{3}}\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(1 - maxCos\right) \cdot \left(ux \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)} \]

FPCore

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

↓

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (cbrt (* (expm1 (log1p (* 8.0 (pow PI 3.0)))) (pow uy 3.0))))
  (sqrt
   (fma
    ux
    (- (- 2.0 maxCos) maxCos)
    (* (- 1.0 maxCos) (* ux (* ux (+ maxCos -1.0))))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf(((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 sinf(cbrtf((expm1f(log1pf((8.0f * powf(((float) M_PI), 3.0f)))) * powf(uy, 3.0f)))) * sqrtf(fmaf(ux, ((2.0f - maxCos) - maxCos), ((1.0f - maxCos) * (ux * (ux * (maxCos + -1.0f))))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(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(sin(cbrt(Float32(expm1(log1p(Float32(Float32(8.0) * (Float32(pi) ^ Float32(3.0))))) * (uy ^ Float32(3.0))))) * sqrt(fma(ux, Float32(Float32(Float32(2.0) - maxCos) - maxCos), Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * Float32(ux * Float32(maxCos + Float32(-1.0))))))))
end

Derivation

1. Initial program 13.5

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

2. Simplified 13.5

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

   [Start] 13.5	\[ \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   associate-*l* [=>] 13.5	\[ \sin \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   cancel-sign-sub-inv [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{1 + \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]
   +-commutative [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + 1}} \]
   *-commutative [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(-\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]
   fma-def [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 - ux\right) + ux \cdot maxCos, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)}} \]
   +-commutative [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{ux \cdot maxCos + \left(1 - ux\right)}, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   associate-+r- [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\left(ux \cdot maxCos + 1\right) - ux}, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   fma-def [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)} - ux, -\left(\left(1 - ux\right) + ux \cdot maxCos\right), 1\right)} \]
   neg-sub0 [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, 1\right)} \]
   +-commutative [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, 1\right)} \]
   associate-+r- [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, 0 - \color{blue}{\left(\left(ux \cdot maxCos + 1\right) - ux\right)}, 1\right)} \]
   associate--r- [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{\left(0 - \left(ux \cdot maxCos + 1\right)\right) + ux}, 1\right)} \]
   +-commutative [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux + \left(0 - \left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]
   sub0-neg [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux + \color{blue}{\left(-\left(ux \cdot maxCos + 1\right)\right)}, 1\right)} \]
   sub-neg [<=] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, \color{blue}{ux - \left(ux \cdot maxCos + 1\right)}, 1\right)} \]
   fma-def [=>] 13.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux, ux - \color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)}, 1\right)} \]

3. Taylor expanded in ux around 0 0.6

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

4. Simplified 0.6

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

   [Start] 0.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right) + ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)} \]
   +-commutative [=>] 0.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right) + \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)}} \]
   fma-def [=>] 0.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(ux, \left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)}} \]
   mul-1-neg [=>] 0.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(1 + \color{blue}{\left(-\left(maxCos - 1\right)\right)}\right) - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   unsub-neg [=>] 0.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \color{blue}{\left(1 - \left(maxCos - 1\right)\right)} - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   associate-+l- [<=] 0.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \color{blue}{\left(\left(1 - maxCos\right) + 1\right)} - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   +-commutative [<=] 0.5	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \color{blue}{\left(1 + \left(1 - maxCos\right)\right)} - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   associate-+r- [=>] 0.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \color{blue}{\left(\left(1 + 1\right) - maxCos\right)} - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   metadata-eval [=>] 0.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(\color{blue}{2} - maxCos\right) - maxCos, \left(maxCos - 1\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   sub-neg [=>] 0.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \color{blue}{\left(maxCos + \left(-1\right)\right)} \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   metadata-eval [=>] 0.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(maxCos + \color{blue}{-1}\right) \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   +-commutative [=>] 0.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \color{blue}{\left(-1 + maxCos\right)} \cdot \left(\left(1 - maxCos\right) \cdot {ux}^{2}\right)\right)} \]
   *-commutative [=>] 0.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(-1 + maxCos\right) \cdot \color{blue}{\left({ux}^{2} \cdot \left(1 - maxCos\right)\right)}\right)} \]
   unpow2 [=>] 0.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(-1 + maxCos\right) \cdot \left(\color{blue}{\left(ux \cdot ux\right)} \cdot \left(1 - maxCos\right)\right)\right)} \]
   associate-*l* [=>] 0.6	\[ \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(-1 + maxCos\right) \cdot \color{blue}{\left(ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)}\right)} \]

5. Applied egg-rr 0.6

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

6. Applied egg-rr 0.6

   \[\leadsto \sin \left(\sqrt[3]{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(8 \cdot {\pi}^{3}\right)\right)} \cdot {uy}^{3}}\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(-1 + maxCos\right) \cdot \left(ux \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

7. Final simplification 0.6

   \[\leadsto \sin \left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left(8 \cdot {\pi}^{3}\right)\right) \cdot {uy}^{3}}\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(1 - maxCos\right) \cdot \left(ux \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)} \]

Alternatives

Alternative 1
Error	0.5
Cost	19968

\[\sqrt[3]{{\left(\mathsf{fma}\left(ux, 2 - \left(maxCos + maxCos\right), \left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot ux\right)\right)\right)\right)}^{1.5} \cdot {\sin \left(uy \cdot \left(\pi \cdot 2\right)\right)}^{3}} \]

Alternative 2
Error	0.6
Cost	13504

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

Alternative 3
Error	0.5
Cost	13472

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

Alternative 4
Error	1.3
Cost	10436

\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0003000000142492354:\\ \;\;\;\;2 \cdot \left(\left(\pi \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(ux, \left(2 - maxCos\right) - maxCos, \left(1 - maxCos\right) \cdot \left(ux \cdot \left(ux \cdot maxCos - ux\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot 2 - ux \cdot ux} \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right)\\ \end{array} \]

Alternative 5
Error	1.3
Cost	10436

\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0003000000142492354:\\ \;\;\;\;\pi \cdot \left(\left(uy \cdot 2\right) \cdot \sqrt{\mathsf{fma}\left(2 + maxCos \cdot -2, ux, \left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot ux\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot 2 - ux \cdot ux} \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right)\\ \end{array} \]

Alternative 6
Error	0.8
Cost	10432

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

Alternative 7
Error	0.8
Cost	10432

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

Alternative 8
Error	4.8
Cost	9988

\[\begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0012000000569969416:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - ux\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot uy\right)\right) \cdot \sqrt{ux + ux}\\ \end{array} \]

Alternative 9
Error	2.5
Cost	9984

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

Alternative 10
Error	2.5
Cost	9920

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

Alternative 11
Error	7.5
Cost	6720

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

Alternative 12
Error	11.8
Cost	6656

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

Reproduce

herbie shell --seed 2023012 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :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)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))