?

Average Error: 19.21% → 0.63%
Time: 30.3s
Precision: binary64

?

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
↓
\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]
(FPCore (cosTheta_i u1 u2)
 :precision binary64
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary64
 (* (sqrt (- (log1p (- u1)))) (cos (* 2.0 (* u2 PI)))))
double code(double cosTheta_i, double u1, double u2) {
	return sqrt(-log((1.0 - u1))) * cos(((2.0 * ((double) M_PI)) * u2));
}
↓
double code(double cosTheta_i, double u1, double u2) {
	return sqrt(-log1p(-u1)) * cos((2.0 * (u2 * ((double) M_PI))));
}

public static double code(double cosTheta_i, double u1, double u2) {
	return Math.sqrt(-Math.log((1.0 - u1))) * Math.cos(((2.0 * Math.PI) * u2));
}
↓
public static double code(double cosTheta_i, double u1, double u2) {
	return Math.sqrt(-Math.log1p(-u1)) * Math.cos((2.0 * (u2 * Math.PI)));
}

def code(cosTheta_i, u1, u2):
	return math.sqrt(-math.log((1.0 - u1))) * math.cos(((2.0 * math.pi) * u2))
↓
def code(cosTheta_i, u1, u2):
	return math.sqrt(-math.log1p(-u1)) * math.cos((2.0 * (u2 * math.pi)))

function code(cosTheta_i, u1, u2)
	return Float64(sqrt(Float64(-log(Float64(1.0 - u1)))) * cos(Float64(Float64(2.0 * pi) * u2)))
end
↓
function code(cosTheta_i, u1, u2)
	return Float64(sqrt(Float64(-log1p(Float64(-u1)))) * cos(Float64(2.0 * Float64(u2 * pi))))
end

code[cosTheta$95$i_, u1_, u2_] := N[(N[Sqrt[(-N[Log[N[(1.0 - u1), $MachinePrecision]], $MachinePrecision])], $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[cosTheta$95$i_, u1_, u2_] := N[(N[Sqrt[(-N[Log[1 + (-u1)], $MachinePrecision])], $MachinePrecision] * N[Cos[N[(2.0 * N[(u2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
↓
\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 19.21

    \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

  2. Simplified0.63

    \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
    Proof

Reproduce?

herbie shell --seed 2023136 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary64
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))