UniformSampleCone 2

Average Error: 0.4 → 0.4

Time: 1.5min

Precision: binary64

Cost: 48512

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

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

↓

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t_0 \cdot t_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t_2 \cdot t_1\right) \cdot xi + \left(\sin t_2 \cdot t_1\right) \cdot yi\right) + \left(\left(ux - {ux}^{2}\right) \cdot maxCos\right) \cdot zi \end{array} \]

FPCore

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

↓

(FPCore (xi yi zi ux uy maxCos)
 :precision binary64
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+
    (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi))
    (* (* (- ux (pow ux 2.0)) maxCos) zi))))

C

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

↓

double code(double xi, double yi, double zi, double ux, double uy, double maxCos) {
	double t_0 = ((1.0 - ux) * maxCos) * ux;
	double t_1 = sqrt((1.0 - (t_0 * t_0)));
	double t_2 = (uy * 2.0) * ((double) M_PI);
	return (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (((ux - pow(ux, 2.0)) * maxCos) * zi);
}

Java

public static double code(double xi, double yi, double zi, double ux, double uy, double maxCos) {
	return (((Math.cos(((uy * 2.0) * Math.PI)) * Math.sqrt((1.0 - ((((1.0 - ux) * maxCos) * ux) * (((1.0 - ux) * maxCos) * ux))))) * xi) + ((Math.sin(((uy * 2.0) * Math.PI)) * Math.sqrt((1.0 - ((((1.0 - ux) * maxCos) * ux) * (((1.0 - ux) * maxCos) * ux))))) * yi)) + ((((1.0 - ux) * maxCos) * ux) * zi);
}

↓

public static double code(double xi, double yi, double zi, double ux, double uy, double maxCos) {
	double t_0 = ((1.0 - ux) * maxCos) * ux;
	double t_1 = Math.sqrt((1.0 - (t_0 * t_0)));
	double t_2 = (uy * 2.0) * Math.PI;
	return (((Math.cos(t_2) * t_1) * xi) + ((Math.sin(t_2) * t_1) * yi)) + (((ux - Math.pow(ux, 2.0)) * maxCos) * zi);
}

Python

def code(xi, yi, zi, ux, uy, maxCos):
	return (((math.cos(((uy * 2.0) * math.pi)) * math.sqrt((1.0 - ((((1.0 - ux) * maxCos) * ux) * (((1.0 - ux) * maxCos) * ux))))) * xi) + ((math.sin(((uy * 2.0) * math.pi)) * math.sqrt((1.0 - ((((1.0 - ux) * maxCos) * ux) * (((1.0 - ux) * maxCos) * ux))))) * yi)) + ((((1.0 - ux) * maxCos) * ux) * zi)

↓

def code(xi, yi, zi, ux, uy, maxCos):
	t_0 = ((1.0 - ux) * maxCos) * ux
	t_1 = math.sqrt((1.0 - (t_0 * t_0)))
	t_2 = (uy * 2.0) * math.pi
	return (((math.cos(t_2) * t_1) * xi) + ((math.sin(t_2) * t_1) * yi)) + (((ux - math.pow(ux, 2.0)) * maxCos) * zi)

Julia

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

↓

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float64(Float64(Float64(1.0 - ux) * maxCos) * ux)
	t_1 = sqrt(Float64(1.0 - Float64(t_0 * t_0)))
	t_2 = Float64(Float64(uy * 2.0) * pi)
	return Float64(Float64(Float64(Float64(cos(t_2) * t_1) * xi) + Float64(Float64(sin(t_2) * t_1) * yi)) + Float64(Float64(Float64(ux - (ux ^ 2.0)) * maxCos) * zi))
end

MATLAB

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

↓

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((1.0 - ux) * maxCos) * ux;
	t_1 = sqrt((1.0 - (t_0 * t_0)));
	t_2 = (uy * 2.0) * pi;
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (((ux - (ux ^ 2.0)) * maxCos) * zi);
end

Wolfram

code[xi_, yi_, zi_, ux_, uy_, maxCos_] := N[(N[(N[(N[(N[Cos[N[(N[(uy * 2.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(N[(N[(1.0 - ux), $MachinePrecision] * maxCos), $MachinePrecision] * ux), $MachinePrecision] * N[(N[(N[(1.0 - ux), $MachinePrecision] * maxCos), $MachinePrecision] * ux), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * xi), $MachinePrecision] + N[(N[(N[Sin[N[(N[(uy * 2.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(N[(N[(1.0 - ux), $MachinePrecision] * maxCos), $MachinePrecision] * ux), $MachinePrecision] * N[(N[(N[(1.0 - ux), $MachinePrecision] * maxCos), $MachinePrecision] * ux), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * yi), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(1.0 - ux), $MachinePrecision] * maxCos), $MachinePrecision] * ux), $MachinePrecision] * zi), $MachinePrecision]), $MachinePrecision]

↓

code[xi_, yi_, zi_, ux_, uy_, maxCos_] := Block[{t$95$0 = N[(N[(N[(1.0 - ux), $MachinePrecision] * maxCos), $MachinePrecision] * ux), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(uy * 2.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[(N[(N[(N[Cos[t$95$2], $MachinePrecision] * t$95$1), $MachinePrecision] * xi), $MachinePrecision] + N[(N[(N[Sin[t$95$2], $MachinePrecision] * t$95$1), $MachinePrecision] * yi), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(ux - N[Power[ux, 2.0], $MachinePrecision]), $MachinePrecision] * maxCos), $MachinePrecision] * zi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 0.4

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

2. Applied egg-rr 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	42176

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t_0 \cdot t_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t_2 \cdot t_1\right) \cdot xi + \left(\sin t_2 \cdot t_1\right) \cdot yi\right) + t_0 \cdot zi \end{array} \]

Alternative 2
Error	0.4
Cost	42176

\[\begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t_0 \cdot t_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t_2 \cdot t_1\right) \cdot xi + \left(\sin t_2 \cdot t_1\right) \cdot yi\right) + \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot zi \end{array} \]

Alternative 3
Error	0.4
Cost	40960

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

Alternative 4
Error	0.4
Cost	34624

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

Alternative 5
Error	0.5
Cost	34368

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

Alternative 6
Error	11.6
Cost	28224

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

Alternative 7
Error	11.6
Cost	28224

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

Alternative 8
Error	61.1
Cost	27328

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

Reproduce

herbie shell --seed 2023033 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary64
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (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) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))