normal distribution

Percentage Accurate: 99.4% → 99.5%

Time: 10.9s

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

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  0.5
  (pow
   (*
    (* (* -2.0 (log u1)) 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((((-2.0 * log(u1)) * 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((((-2.0 * Math.log(u1)) * 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((((-2.0 * math.log(u1)) * 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(-2.0 * log(u1)) * 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 + ((((-2.0 * log(u1)) * 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[(-2.0 * N[Log[u1], $MachinePrecision]), $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 99.5%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. 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 rewrites 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 rewrites 99.7%

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

7. Final simplification 99.7%

   \[\leadsto 0.5 + {\left(\left(\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)\right)}^{0.5} \]
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 99.5%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. 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 rewrites 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 rewrites 99.7%

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

8. Applied rewrites 99.7%

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

Alternative 3: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 0.5 + {\left(\log u1 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 0.2222222222222222, \pi \cdot \pi, -0.05555555555555555\right)\right)}^{0.5} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  0.5
  (pow
   (*
    (log u1)
    (fma (* (* u2 u2) 0.2222222222222222) (* PI PI) -0.05555555555555555))
   0.5)))

C

double code(double u1, double u2) {
	return 0.5 + pow((log(u1) * fma(((u2 * u2) * 0.2222222222222222), (((double) M_PI) * ((double) M_PI)), -0.05555555555555555)), 0.5);
}

Julia

function code(u1, u2)
	return Float64(0.5 + (Float64(log(u1) * fma(Float64(Float64(u2 * u2) * 0.2222222222222222), Float64(pi * pi), -0.05555555555555555)) ^ 0.5))
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[Power[N[(N[Log[u1], $MachinePrecision] * N[(N[(N[(u2 * u2), $MachinePrecision] * 0.2222222222222222), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + -0.05555555555555555), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. 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 rewrites 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 rewrites 99.7%

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

7. Taylor expanded in u2 around 0

   \[\leadsto {\color{blue}{\left(\frac{-1}{18} \cdot \log u1 + \frac{2}{9} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \log u1\right)\right)\right)}}^{\frac{1}{2}} + \frac{1}{2} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto {\color{blue}{\left(\frac{2}{9} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \log u1\right)\right) + \frac{-1}{18} \cdot \log u1\right)}}^{\frac{1}{2}} + \frac{1}{2} \]

   2. associate-*r* N/A

      \[\leadsto {\left(\color{blue}{\left(\frac{2}{9} \cdot {u2}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \log u1\right)} + \frac{-1}{18} \cdot \log u1\right)}^{\frac{1}{2}} + \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto {\left(\color{blue}{\left(\left(\frac{2}{9} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \log u1} + \frac{-1}{18} \cdot \log u1\right)}^{\frac{1}{2}} + \frac{1}{2} \]

   4. distribute-rgt-out N/A

      \[\leadsto {\color{blue}{\left(\log u1 \cdot \left(\left(\frac{2}{9} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{18}\right)\right)}}^{\frac{1}{2}} + \frac{1}{2} \]

   5. lower-*.f64 N/A

      \[\leadsto {\color{blue}{\left(\log u1 \cdot \left(\left(\frac{2}{9} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{18}\right)\right)}}^{\frac{1}{2}} + \frac{1}{2} \]

   6. lower-log.f64 N/A

      \[\leadsto {\left(\color{blue}{\log u1} \cdot \left(\left(\frac{2}{9} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{-1}{18}\right)\right)}^{\frac{1}{2}} + \frac{1}{2} \]

   7. lower-fma.f64 N/A

      \[\leadsto {\left(\log u1 \cdot \color{blue}{\mathsf{fma}\left(\frac{2}{9} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{2}, \frac{-1}{18}\right)}\right)}^{\frac{1}{2}} + \frac{1}{2} \]

   8. *-commutative N/A

      \[\leadsto {\left(\log u1 \cdot \mathsf{fma}\left(\color{blue}{{u2}^{2} \cdot \frac{2}{9}}, {\mathsf{PI}\left(\right)}^{2}, \frac{-1}{18}\right)\right)}^{\frac{1}{2}} + \frac{1}{2} \]

   9. lower-*.f64 N/A

      \[\leadsto {\left(\log u1 \cdot \mathsf{fma}\left(\color{blue}{{u2}^{2} \cdot \frac{2}{9}}, {\mathsf{PI}\left(\right)}^{2}, \frac{-1}{18}\right)\right)}^{\frac{1}{2}} + \frac{1}{2} \]

   10. unpow2 N/A

       \[\leadsto {\left(\log u1 \cdot \mathsf{fma}\left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{2}{9}, {\mathsf{PI}\left(\right)}^{2}, \frac{-1}{18}\right)\right)}^{\frac{1}{2}} + \frac{1}{2} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(\log u1 \cdot \mathsf{fma}\left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{2}{9}, {\mathsf{PI}\left(\right)}^{2}, \frac{-1}{18}\right)\right)}^{\frac{1}{2}} + \frac{1}{2} \]

   12. unpow2 N/A

       \[\leadsto {\left(\log u1 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{2}{9}, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{-1}{18}\right)\right)}^{\frac{1}{2}} + \frac{1}{2} \]

   13. lower-*.f64 N/A

       \[\leadsto {\left(\log u1 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{2}{9}, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{-1}{18}\right)\right)}^{\frac{1}{2}} + \frac{1}{2} \]

   14. lower-PI.f64 N/A

       \[\leadsto {\left(\log u1 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{2}{9}, \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), \frac{-1}{18}\right)\right)}^{\frac{1}{2}} + \frac{1}{2} \]

   15. lower-PI.f64 99.6

       \[\leadsto {\left(\log u1 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 0.2222222222222222, \pi \cdot \color{blue}{\pi}, -0.05555555555555555\right)\right)}^{0.5} + 0.5 \]

9. Applied rewrites 99.6%

   \[\leadsto {\color{blue}{\left(\log u1 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 0.2222222222222222, \pi \cdot \pi, -0.05555555555555555\right)\right)}}^{0.5} + 0.5 \]

10. Final simplification 99.6%

    \[\leadsto 0.5 + {\left(\log u1 \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot 0.2222222222222222, \pi \cdot \pi, -0.05555555555555555\right)\right)}^{0.5} \]
11. Add Preprocessing

Alternative 4: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double u1, double u2) {
	return 0.5 + (((1.0 / 6.0) * (sqrt(-log(u1)) * sqrt(2.0))) * fma(u2, (u2 * (-2.0 * (((double) M_PI) * ((double) M_PI)))), 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(u2 * Float64(-2.0 * Float64(pi * pi))), 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[(u2 * N[(-2.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
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 99.5

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

5. Applied rewrites 99.5%

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

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

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

   3. rem-square-sqrt N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. unpow2 N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

8. Applied rewrites 99.4%

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

9. Final simplification 99.4%

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

Alternative 5: 98.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{-\log u1 \cdot 2} \cdot \mathsf{fma}\left(u2, -2 \cdot \left(u2 \cdot \left(\pi \cdot \pi\right)\right), 1\right), 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (sqrt (- (* (log u1) 2.0))) (fma u2 (* -2.0 (* u2 (* PI PI))) 1.0))
  0.16666666666666666
  0.5))

C

double code(double u1, double u2) {
	return fma((sqrt(-(log(u1) * 2.0)) * fma(u2, (-2.0 * (u2 * (((double) M_PI) * ((double) M_PI)))), 1.0)), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(sqrt(Float64(-Float64(log(u1) * 2.0))) * fma(u2, Float64(-2.0 * Float64(u2 * Float64(pi * pi))), 1.0)), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[(-N[(N[Log[u1], $MachinePrecision] * 2.0), $MachinePrecision])], $MachinePrecision] * N[(u2 * N[(-2.0 * N[(u2 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
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 99.5

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

5. Applied rewrites 99.5%

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

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

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

   3. rem-square-sqrt N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. unpow2 N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

8. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

10. Applied rewrites 99.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-\log u1 \cdot 2} \cdot \mathsf{fma}\left(u2, -2 \cdot \left(u2 \cdot \left(\pi \cdot \pi\right)\right), 1\right), 0.16666666666666666, 0.5\right)} \]
11. Add Preprocessing

Alternative 6: 98.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\log u1 \cdot -0.05555555555555555} - -0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (- (sqrt (* (log u1) -0.05555555555555555)) -0.5))

C

double code(double u1, double u2) {
	return sqrt((log(u1) * -0.05555555555555555)) - -0.5;
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = sqrt((log(u1) * (-0.05555555555555555d0))) - (-0.5d0)
end function

Java

public static double code(double u1, double u2) {
	return Math.sqrt((Math.log(u1) * -0.05555555555555555)) - -0.5;
}

Python

def code(u1, u2):
	return math.sqrt((math.log(u1) * -0.05555555555555555)) - -0.5

Julia

function code(u1, u2)
	return Float64(sqrt(Float64(log(u1) * -0.05555555555555555)) - -0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = sqrt((log(u1) * -0.05555555555555555)) - -0.5;
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision] - -0.5), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
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. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 0.0

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

5. Applied rewrites 0.0%

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

7. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, \color{blue}{0.16666666666666666}, 0.5\right) \]
8. Step-by-step derivation

9. Applied rewrites 99.3%

   \[\leadsto \sqrt{\log u1 \cdot -0.05555555555555555} - \color{blue}{-0.5} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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))