normal distribution

Percentage Accurate: 99.4% → 99.5%

Time: 11.5s

Alternatives: 6

Speedup: 1.5×

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, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + {\left(\left(\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  0.5
  (pow
   (*
    (* (* (log u1) -2.0) 0.027777777777777776)
    (+ 0.5 (* 0.5 (cos (* 2.0 (* 2.0 (* PI u2)))))))
   0.5)))

C

double code(double u1, double u2) {
	return 0.5 + pow((((log(u1) * -2.0) * 0.027777777777777776) * (0.5 + (0.5 * cos((2.0 * (2.0 * (((double) M_PI) * u2))))))), 0.5);
}

Java

public static double code(double u1, double u2) {
	return 0.5 + Math.pow((((Math.log(u1) * -2.0) * 0.027777777777777776) * (0.5 + (0.5 * Math.cos((2.0 * (2.0 * (Math.PI * u2))))))), 0.5);
}

Python

def code(u1, u2):
	return 0.5 + math.pow((((math.log(u1) * -2.0) * 0.027777777777777776) * (0.5 + (0.5 * math.cos((2.0 * (2.0 * (math.pi * u2))))))), 0.5)

Julia

function code(u1, u2)
	return Float64(0.5 + (Float64(Float64(Float64(log(u1) * -2.0) * 0.027777777777777776) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(2.0 * Float64(pi * u2))))))) ^ 0.5))
end

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + ((((log(u1) * -2.0) * 0.027777777777777776) * (0.5 + (0.5 * cos((2.0 * (2.0 * (pi * u2))))))) ^ 0.5);
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[Power[N[(N[(N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision] * 0.027777777777777776), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(2.0 * N[(Pi * u2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $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. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{\frac{1}{4}}}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   7. metadata-eval N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval 99.0

      \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied egg-rr 99.0%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left({\left(-2 \cdot \log u1\right)}^{0.25}\right)}^{2}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(\left(\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)\right)}^{0.5}} + 0.5 \]

7. Final simplification 99.6%

   \[\leadsto 0.5 + {\left(\left(\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)\right)}^{0.5} \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \left|\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right|, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (sqrt (* (log u1) -0.05555555555555555))
  (fabs (cos (* 2.0 (* PI u2))))
  0.5))

C

double code(double u1, double u2) {
	return fma(sqrt((log(u1) * -0.05555555555555555)), fabs(cos((2.0 * (((double) M_PI) * u2)))), 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(Float64(log(u1) * -0.05555555555555555)), abs(cos(Float64(2.0 * Float64(pi * u2)))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision] * N[Abs[N[Cos[N[(2.0 * N[(Pi * u2), $MachinePrecision]), $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-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{\frac{1}{4}}}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   7. metadata-eval N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval 99.1

      \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{0.25}}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied egg-rr 99.1%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left({\left(-2 \cdot \log u1\right)}^{0.25}\right)}^{2}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{{\left(\left(\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)\right)}^{0.5}} + 0.5 \]

8. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \left|\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right|, 0.5\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(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))