Beckmann Sample, near normal, slope_x

Average Error: 12.5 → 0.4

Time: 22.9s

Precision: binary64

Cost: 26112

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

↓

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(u2 + u2\right) \cdot \pi\right) \]

FPCore

(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 (* (+ u2 u2) PI))))

C

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(((u2 + u2) * ((double) M_PI)));
}

Java

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(((u2 + u2) * Math.PI));
}

Python

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(((u2 + u2) * math.pi))

Julia

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(Float64(u2 + u2) * pi)))
end

Wolfram

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[(N[(u2 + u2), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 12.5

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

2. Simplified 0.4

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

3. Applied egg-rr 0.4

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

Alternatives

Alternative 1
Error	21.6
Cost	32968

\[\begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ t_1 := \mathsf{log1p}\left(-u1\right)\\ \mathbf{if}\;t_0 \leq 0.999994:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\begin{array}{l} \mathbf{if}\;t_1 \ne 0:\\ \;\;\;\;\frac{-1}{\frac{-1}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}}\\ \end{array} \]

Alternative 2
Error	21.6
Cost	32836

\[\begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t_0 \leq 0.999994:\\ \;\;\;\;\sqrt{-\left(-u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

Alternative 3
Error	26.0
Cost	12992

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \]

Alternative 4
Error	30.6
Cost	7296

\[\sqrt{-\left(u1 \cdot \left(-0.3333333333333333 \cdot \left(u1 \cdot u1\right) + -0.5 \cdot u1\right) - u1\right)} \]

Alternative 5
Error	30.6
Cost	7104

\[\sqrt{u1 - \left(u1 \cdot u1\right) \cdot \left(-0.3333333333333333 \cdot u1 + -0.5\right)} \]

Alternative 6
Error	33.0
Cost	6912

\[\sqrt{-\left(-0.5 \cdot \left(u1 \cdot u1\right) - u1\right)} \]

Alternative 7
Error	39.5
Cost	6592

\[\sqrt{-\left(-u1\right)} \]

Reproduce

herbie shell --seed 2023010 
(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))))