UniformSampleCone, y

Average Error: 13.6 → 0.5

Time: 30.5s

Precision: binary32

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

Math

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

↓

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

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
 (expm1
  (log1p
   (*
    (sqrt
     (-
      (* 2.0 (fma maxCos (* ux ux) ux))
      (fma maxCos (* ux (fma maxCos ux 2.0)) (* ux ux))))
    (sin (* 2.0 (* PI uy)))))))

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 expm1f(log1pf(sqrtf((2.0f * fmaf(maxCos, (ux * ux), ux)) - fmaf(maxCos, (ux * fmaf(maxCos, ux, 2.0f)), (ux * ux))) * sinf(2.0f * (((float) M_PI) * uy))));
}

Error

Bits error versus ux

Bits error versus uy

Bits error versus maxCos

Derivation

1. Initial program 13.6

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

2. Taylor expanded in ux around 0 0.5

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

3. Simplified 0.5

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

4. Applied expm1-log1p-u_binary32 0.5

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

5. Simplified 0.5

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

6. Final simplification 0.5

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

Reproduce

herbie shell --seed 2021275 
(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)))))))