normal distribution

Percentage Accurate: 99.4% → 99.5%

Time: 12.4s

Alternatives: 8

Speedup: 1.4×

Specification

\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Alternative 1: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 0.5 + \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  0.5
  (*
   (* (/ 1.0 6.0) (* (sqrt (- (log u1))) (sqrt 2.0)))
   (cos (* (* 2.0 PI) u2)))))

C

double code(double u1, double u2) {
	return 0.5 + (((1.0 / 6.0) * (sqrt(-log(u1)) * sqrt(2.0))) * cos(((2.0 * ((double) M_PI)) * u2)));
}

Java

public static double code(double u1, double u2) {
	return 0.5 + (((1.0 / 6.0) * (Math.sqrt(-Math.log(u1)) * Math.sqrt(2.0))) * Math.cos(((2.0 * Math.PI) * u2)));
}

Python

def code(u1, u2):
	return 0.5 + (((1.0 / 6.0) * (math.sqrt(-math.log(u1)) * math.sqrt(2.0))) * math.cos(((2.0 * math.pi) * u2)))

Julia

function code(u1, u2)
	return Float64(0.5 + Float64(Float64(Float64(1.0 / 6.0) * Float64(sqrt(Float64(-log(u1))) * sqrt(2.0))) * cos(Float64(Float64(2.0 * pi) * u2))))
end

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + (((1.0 / 6.0) * (sqrt(-log(u1)) * sqrt(2.0))) * cos(((2.0 * pi) * u2)));
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[(N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lower-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. lower-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. lower-sqrt.f64 99.4

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Applied rewrites 99.4%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Final simplification 99.4%

   \[\leadsto 0.5 + \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
7. Add Preprocessing

Alternative 2: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{-2 \cdot \log u1}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* 0.16666666666666666 (cos (* 2.0 (* PI u2))))
  (sqrt (* -2.0 (log u1)))
  0.5))

C

double code(double u1, double u2) {
	return fma((0.16666666666666666 * cos((2.0 * (((double) M_PI) * u2)))), sqrt((-2.0 * log(u1))), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(0.16666666666666666 * cos(Float64(2.0 * Float64(pi * u2)))), sqrt(Float64(-2.0 * log(u1))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(0.16666666666666666 * N[Cos[N[(2.0 * N[(Pi * u2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} + \frac{1}{2} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({\left(-2 \cdot \log u1\right)}^{\frac{1}{2}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} + \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right)} + \frac{1}{2} \]

   6. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}} + \frac{1}{2} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right), {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}, \frac{1}{2}\right)} \]

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{-2 \cdot \log u1}, 0.5\right)} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (and (<= 0.0 u1) (<= u1 1.0)) (and (<= 0.0 u2) (<= u2 1.0)))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))