normal distribution

Percentage Accurate: 99.4% → 99.4%

Time: 6.8s

Alternatives: 6

Speedup: 1.3×

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[u1_, u2_] := N[(0.5 + 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]), $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. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 3: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u1, u2)
	return Float64(0.5 + Float64(Float64(Float64(1.0 / 6.0) * Float64(sqrt(Float64(-log(u1))) * sqrt(2.0))) * fma(u2, Float64(-2.0 * Float64(pi * Float64(pi * u2))), 1.0)))
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[(u2 * N[(-2.0 * N[(Pi * N[(Pi * u2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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. Simplified 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. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-PI.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-PI.f64 98.9

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 4: 98.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.16666666666666666), $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. 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. Simplified 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. Taylor expanded in u2 around 0

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

8. Simplified 98.4%

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

9. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right)} \cdot 1 + \frac{1}{2} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. associate-*l* N/A

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

    3. lower-*.f64 N/A

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

    4. lower-sqrt.f64 N/A

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

    5. log-rec N/A

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

    6. lower-neg.f64 N/A

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

    7. lower-log.f64 N/A

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

    8. *-commutative N/A

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

    9. unpow2 N/A

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

    10. rem-square-sqrt N/A

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

    11. associate-*l* N/A

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

    12. metadata-eval N/A

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

    13. metadata-eval N/A

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

    14. lower-*.f64 N/A

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

    15. lower-sqrt.f64 N/A

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

    16. metadata-eval 1.7

        \[\leadsto \left(\sqrt{-\log u1} \cdot \left(\sqrt{2} \cdot \color{blue}{-0.16666666666666666}\right)\right) \cdot 1 + 0.5 \]

11. Simplified 1.7%

    \[\leadsto \color{blue}{\left(\sqrt{-\log u1} \cdot \left(\sqrt{2} \cdot -0.16666666666666666\right)\right)} \cdot 1 + 0.5 \]

12. Taylor expanded in u1 around inf

    \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{6} \cdot \left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
13. Step-by-step derivation

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-sqrt.f64 N/A

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

    6. log-rec N/A

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

    7. lower-neg.f64 N/A

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

    8. lower-log.f64 N/A

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

    9. *-commutative N/A

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

    10. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot \frac{-1}{6}, \frac{1}{2}\right) \]

    11. rem-square-sqrt N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \left(\sqrt{2} \cdot \color{blue}{-1}\right) \cdot \frac{-1}{6}, \frac{1}{2}\right) \]

    12. associate-*l* N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \color{blue}{\sqrt{2} \cdot \left(-1 \cdot \frac{-1}{6}\right)}, \frac{1}{2}\right) \]

    13. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \sqrt{2} \cdot \color{blue}{\frac{1}{6}}, \frac{1}{2}\right) \]

    14. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \sqrt{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot -1\right)}, \frac{1}{2}\right) \]

    15. lower-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \color{blue}{\sqrt{2} \cdot \left(\frac{-1}{6} \cdot -1\right)}, \frac{1}{2}\right) \]

    16. lower-sqrt.f64 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \color{blue}{\sqrt{2}} \cdot \left(\frac{-1}{6} \cdot -1\right), \frac{1}{2}\right) \]

    17. metadata-eval 98.5

        \[\leadsto \mathsf{fma}\left(\sqrt{-\log u1}, \sqrt{2} \cdot \color{blue}{0.16666666666666666}, 0.5\right) \]

14. Simplified 98.5%

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

Alternative 5: 1.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * -0.16666666666666666), $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. 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. Simplified 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. Taylor expanded in u2 around 0

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

8. Simplified 98.4%

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

9. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-sqrt.f64 N/A

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

    6. log-rec N/A

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

    7. lower-neg.f64 N/A

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

    8. lower-log.f64 N/A

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

    9. *-commutative N/A

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

    10. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot \frac{1}{6}, \frac{1}{2}\right) \]

    11. rem-square-sqrt N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \left(\sqrt{2} \cdot \color{blue}{-1}\right) \cdot \frac{1}{6}, \frac{1}{2}\right) \]

    12. associate-*l* N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \color{blue}{\sqrt{2} \cdot \left(-1 \cdot \frac{1}{6}\right)}, \frac{1}{2}\right) \]

    13. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \sqrt{2} \cdot \color{blue}{\frac{-1}{6}}, \frac{1}{2}\right) \]

    14. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \sqrt{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot -1\right)}, \frac{1}{2}\right) \]

    15. lower-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \color{blue}{\sqrt{2} \cdot \left(\frac{1}{6} \cdot -1\right)}, \frac{1}{2}\right) \]

    16. lower-sqrt.f64 N/A

        \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{neg}\left(\log u1\right)}, \color{blue}{\sqrt{2}} \cdot \left(\frac{1}{6} \cdot -1\right), \frac{1}{2}\right) \]

    17. metadata-eval 1.7

        \[\leadsto \mathsf{fma}\left(\sqrt{-\log u1}, \sqrt{2} \cdot \color{blue}{-0.16666666666666666}, 0.5\right) \]

11. Simplified 1.7%

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

Alternative 6: 0.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[Log[u1], $MachinePrecision]], $MachinePrecision] * N[(0.16666666666666666 * N[Sqrt[-2.0], $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. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 0.0

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, 0.16666666666666666 \cdot \color{blue}{\sqrt{-2}}, 0.5\right) \]

5. Simplified 0.0%

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

Reproduce

herbie shell --seed 2024215 
(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))