normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 5.3s

Alternatives: 5

Speedup: N/A×

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: 99.6% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.5%

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

Alternative 3: 99.4% accurate, N/A× speedup

Math

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  0.16666666666666666
  (* (pow (* (log u1) -2.0) 0.5) (sin (fma (* PI 2.0) u2 (/ PI 2.0))))
  0.5))

C

double code(double u1, double u2) {
	return fma(0.16666666666666666, (pow((log(u1) * -2.0), 0.5) * sin(fma((((double) M_PI) * 2.0), u2, (((double) M_PI) / 2.0)))), 0.5);
}

Julia

function code(u1, u2)
	return fma(0.16666666666666666, Float64((Float64(log(u1) * -2.0) ^ 0.5) * sin(fma(Float64(pi * 2.0), u2, Float64(pi / 2.0)))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(0.16666666666666666 * N[(N[Power[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision], 0.5], $MachinePrecision] * N[Sin[N[(N[(Pi * 2.0), $MachinePrecision] * u2 + N[(Pi / 2.0), $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-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. pow-to-exp N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-log.f64 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, {\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sin \left(\mathsf{fma}\left(\pi \cdot 2, u2, \frac{\pi}{2}\right)\right), 0.5\right)} \]
7. Add Preprocessing

Alternative 4: 99.4% accurate, N/A× 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]

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. Add Preprocessing

Alternative 5: 99.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\log u1 \cdot -2\right)}^{0.5}\\ t_1 := {t\_0}^{0.25}\\ \left(\left(t\_1 \cdot t\_1\right) \cdot \left({t\_0}^{0.5} \cdot 0.16666666666666666\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (let* ((t_0 (pow (* (log u1) -2.0) 0.5)) (t_1 (pow t_0 0.25)))
   (+
    (*
     (* (* t_1 t_1) (* (pow t_0 0.5) 0.16666666666666666))
     (cos (* (* 2.0 PI) u2)))
    0.5)))

C

double code(double u1, double u2) {
	double t_0 = pow((log(u1) * -2.0), 0.5);
	double t_1 = pow(t_0, 0.25);
	return (((t_1 * t_1) * (pow(t_0, 0.5) * 0.16666666666666666)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	double t_0 = Math.pow((Math.log(u1) * -2.0), 0.5);
	double t_1 = Math.pow(t_0, 0.25);
	return (((t_1 * t_1) * (Math.pow(t_0, 0.5) * 0.16666666666666666)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	t_0 = math.pow((math.log(u1) * -2.0), 0.5)
	t_1 = math.pow(t_0, 0.25)
	return (((t_1 * t_1) * (math.pow(t_0, 0.5) * 0.16666666666666666)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	t_0 = Float64(log(u1) * -2.0) ^ 0.5
	t_1 = t_0 ^ 0.25
	return Float64(Float64(Float64(Float64(t_1 * t_1) * Float64((t_0 ^ 0.5) * 0.16666666666666666)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	t_0 = (log(u1) * -2.0) ^ 0.5;
	t_1 = t_0 ^ 0.25;
	tmp = (((t_1 * t_1) * ((t_0 ^ 0.5) * 0.16666666666666666)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := Block[{t$95$0 = N[Power[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 0.25], $MachinePrecision]}, N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(N[Power[t$95$0, 0.5], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $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

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

   2. lift-/.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. *-commutative N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. pow-unpow N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lift-log.f64 N/A

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

   18. lower-*.f64 N/A

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

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\left({\left({\left(\log u1 \cdot -2\right)}^{0.5}\right)}^{0.5} \cdot \left({\left({\left(\log u1 \cdot -2\right)}^{0.5}\right)}^{0.5} \cdot 0.16666666666666666\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. sqr-pow N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-pow.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-pow.f64 N/A

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

   14. lift-log.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-pow.f64 98.9

       \[\leadsto \left(\left({\left({\left(\log u1 \cdot -2\right)}^{0.5}\right)}^{0.25} \cdot {\color{blue}{\left({\left(\log u1 \cdot -2\right)}^{0.5}\right)}}^{0.25}\right) \cdot \left({\left({\left(\log u1 \cdot -2\right)}^{0.5}\right)}^{0.5} \cdot 0.16666666666666666\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Applied rewrites 98.9%

   \[\leadsto \left(\color{blue}{\left({\left({\left(\log u1 \cdot -2\right)}^{0.5}\right)}^{0.25} \cdot {\left({\left(\log u1 \cdot -2\right)}^{0.5}\right)}^{0.25}\right)} \cdot \left({\left({\left(\log u1 \cdot -2\right)}^{0.5}\right)}^{0.5} \cdot 0.16666666666666666\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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))