UniformSampleCone, x

Average Error: 12.5 → 0.4

Time: 1.8min

Precision: binary64

Cost: 26752

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

Math

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

↓

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary64
 (*
  (cos (* (* uy 2.0) PI))
  (sqrt
   (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))

↓

(FPCore (ux uy maxCos)
 :precision binary64
 (*
  (cos (* (* uy 2.0) PI))
  (sqrt (* (- 1.0 maxCos) (fma (- maxCos 1.0) (* ux ux) (* 2.0 ux))))))

C

double code(double ux, double uy, double maxCos) {
	return cos(((uy * 2.0) * ((double) M_PI))) * sqrt((1.0 - (((1.0 - ux) + (ux * maxCos)) * ((1.0 - ux) + (ux * maxCos)))));
}

↓

double code(double ux, double uy, double maxCos) {
	return cos(((uy * 2.0) * ((double) M_PI))) * sqrt(((1.0 - maxCos) * fma((maxCos - 1.0), (ux * ux), (2.0 * ux))));
}

Julia

function code(ux, uy, maxCos)
	return Float64(cos(Float64(Float64(uy * 2.0) * pi)) * sqrt(Float64(1.0 - Float64(Float64(Float64(1.0 - ux) + Float64(ux * maxCos)) * Float64(Float64(1.0 - ux) + Float64(ux * maxCos))))))
end

↓

function code(ux, uy, maxCos)
	return Float64(cos(Float64(Float64(uy * 2.0) * pi)) * sqrt(Float64(Float64(1.0 - maxCos) * fma(Float64(maxCos - 1.0), Float64(ux * ux), Float64(2.0 * ux)))))
end

Wolfram

code[ux_, uy_, maxCos_] := N[(N[Cos[N[(N[(uy * 2.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(N[(1.0 - ux), $MachinePrecision] + N[(ux * maxCos), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - ux), $MachinePrecision] + N[(ux * maxCos), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[ux_, uy_, maxCos_] := N[(N[Cos[N[(N[(uy * 2.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(1.0 - maxCos), $MachinePrecision] * N[(N[(maxCos - 1.0), $MachinePrecision] * N[(ux * ux), $MachinePrecision] + N[(2.0 * ux), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 12.5

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

2. Applied egg-rr 12.4

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

3. Taylor expanded in ux around 0 0.4

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

4. Simplified 0.4

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

5. Applied egg-rr 0.4

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

6. Taylor expanded in ux around 0 0.4

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

7. Simplified 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	20352

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

Alternative 2
Error	0.4
Cost	20352

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

Alternative 3
Error	1.0
Cost	20096

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

Alternative 4
Error	1.6
Cost	19840

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

Alternative 5
Error	25.9
Cost	13696

\[\sqrt{-\left(\mathsf{fma}\left(ux, maxCos, 1\right) - \left(ux - 1\right)\right) \cdot \left(ux \cdot maxCos - ux\right)} \]

Alternative 6
Error	25.9
Cost	13632

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

Alternative 7
Error	25.9
Cost	7232

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

Alternative 8
Error	25.9
Cost	7232

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

Alternative 9
Error	25.9
Cost	7232

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

Alternative 10
Error	26.3
Cost	6912

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

Alternative 11
Error	39.6
Cost	6848

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

Alternative 12
Error	39.7
Cost	6592

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

Reproduce

herbie shell --seed 2023033 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary64
  :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)))))))