normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 11.8s

Alternatives: 11

Speedup: 0.8×

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, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(u1, u2):
	return ((math.sqrt(-math.log(u1)) * (math.sqrt(2.0) * 0.16666666666666666)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

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

MATLAB

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

Wolfram

code[u1_, u2_] := N[(N[(N[(N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * N[(N[Sqrt[2.0], $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. Taylor expanded in u1 around inf 99.6%

   \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
3. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right) \cdot 0.16666666666666666\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. associate-*l* 99.7%

      \[\leadsto \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(\sqrt{2} \cdot 0.16666666666666666\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   3. log-rec 99.7%

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

4. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.4%

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

   2. fma-def 99.4%

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

   3. cos-neg 99.4%

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

   4. distribute-rgt-neg-out 99.4%

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

   5. metadata-eval 99.4%

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

   6. unpow1/2 99.4%

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

   7. distribute-rgt-neg-out 99.4%

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

   8. cos-neg 99.4%

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

   9. associate-*l* 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(u1, u2):
	return 0.5 + (math.cos(((2.0 * math.pi) * u2)) * (0.16666666666666666 * math.sqrt((math.log(u1) * -2.0))))

Julia

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

MATLAB

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

Wolfram

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

   1. log1p-expm1-u_binary64 99.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5} \]

3. Applied rewrite-once 99.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
4. Step-by-step derivation

   1. log1p-expm1 99.4%

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

   2. metadata-eval 99.4%

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

   3. unpow1/2 99.4%

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

   4. *-commutative 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 98.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = (sqrt(-log(u1)) * (sqrt(2.0d0) * 0.16666666666666666d0)) + 0.5d0
end function

Java

public static double code(double u1, double u2) {
	return (Math.sqrt(-Math.log(u1)) * (Math.sqrt(2.0) * 0.16666666666666666)) + 0.5;
}

Python

def code(u1, u2):
	return (math.sqrt(-math.log(u1)) * (math.sqrt(2.0) * 0.16666666666666666)) + 0.5

Julia

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

MATLAB

function tmp = code(u1, u2)
	tmp = (sqrt(-log(u1)) * (sqrt(2.0) * 0.16666666666666666)) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.16666666666666666), $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. Taylor expanded in u1 around inf 99.6%

   \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
3. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto \color{blue}{\left(\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right) \cdot 0.16666666666666666\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. associate-*l* 99.7%

      \[\leadsto \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(\sqrt{2} \cdot 0.16666666666666666\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   3. log-rec 99.7%

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

4. Simplified 99.7%

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

5. Taylor expanded in u2 around 0 97.8%

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

6. Final simplification 97.8%

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

Alternative 5: 21.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot 4 \end{array} \]

FPCore

(FPCore (u1 u2) :precision binary64 (+ 0.5 (* (cos (* (* 2.0 PI) u2)) 4.0)))

C

double code(double u1, double u2) {
	return 0.5 + (cos(((2.0 * ((double) M_PI)) * u2)) * 4.0);
}

Java

public static double code(double u1, double u2) {
	return 0.5 + (Math.cos(((2.0 * Math.PI) * u2)) * 4.0);
}

Python

def code(u1, u2):
	return 0.5 + (math.cos(((2.0 * math.pi) * u2)) * 4.0)

Julia

function code(u1, u2)
	return Float64(0.5 + Float64(cos(Float64(Float64(2.0 * pi) * u2)) * 4.0))
end

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + (cos(((2.0 * pi) * u2)) * 4.0);
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Applied egg-rr 21.8%

   \[\leadsto \color{blue}{4} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Final simplification 21.8%

   \[\leadsto 0.5 + \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot 4 \]

Alternative 6: 98.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + 0.16666666666666666 \cdot \sqrt{\log u1 \cdot -2} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = 0.5d0 + (0.16666666666666666d0 * sqrt((log(u1) * (-2.0d0))))
end function

Java

public static double code(double u1, double u2) {
	return 0.5 + (0.16666666666666666 * Math.sqrt((Math.log(u1) * -2.0)));
}

Python

def code(u1, u2):
	return 0.5 + (0.16666666666666666 * math.sqrt((math.log(u1) * -2.0)))

Julia

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

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + (0.16666666666666666 * sqrt((log(u1) * -2.0)));
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[(0.16666666666666666 * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $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. Step-by-step derivation

   1. log1p-expm1-u_binary64 99.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5} \]

3. Applied rewrite-once 99.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
4. Step-by-step derivation

   1. log1p-expm1 99.4%

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

   2. metadata-eval 99.4%

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

   3. unpow1/2 99.4%

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

   4. *-commutative 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in u2 around 0 97.5%

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

7. Final simplification 97.5%

   \[\leadsto 0.5 + 0.16666666666666666 \cdot \sqrt{\log u1 \cdot -2} \]

Alternative 7: 17.6% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ 1.25 \end{array} \]

FPCore

(FPCore (u1 u2) :precision binary64 1.25)

C

double code(double u1, double u2) {
	return 1.25;
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = 1.25d0
end function

Java

public static double code(double u1, double u2) {
	return 1.25;
}

Python

def code(u1, u2):
	return 1.25

Julia

function code(u1, u2)
	return 1.25
end

MATLAB

function tmp = code(u1, u2)
	tmp = 1.25;
end

Wolfram

code[u1_, u2_] := 1.25

Derivation

1. Initial program 99.4%

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

3. Applied egg-rr 17.5%

   \[\leadsto \color{blue}{0.75} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Taylor expanded in u2 around 0 17.5%

   \[\leadsto \color{blue}{1.25} \]

5. Final simplification 17.5%

   \[\leadsto 1.25 \]

Alternative 8: 18.0% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ 1.5 \end{array} \]

FPCore

(FPCore (u1 u2) :precision binary64 1.5)

C

double code(double u1, double u2) {
	return 1.5;
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = 1.5d0
end function

Java

public static double code(double u1, double u2) {
	return 1.5;
}

Python

def code(u1, u2):
	return 1.5

Julia

function code(u1, u2)
	return 1.5
end

MATLAB

function tmp = code(u1, u2)
	tmp = 1.5;
end

Wolfram

code[u1_, u2_] := 1.5

Derivation

1. Initial program 99.4%

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

3. Applied egg-rr 17.9%

   \[\leadsto \color{blue}{1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Taylor expanded in u2 around 0 17.9%

   \[\leadsto \color{blue}{1.5} \]

5. Final simplification 17.9%

   \[\leadsto 1.5 \]

Alternative 9: 18.2% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ 1.625 \end{array} \]

FPCore

(FPCore (u1 u2) :precision binary64 1.625)

C

double code(double u1, double u2) {
	return 1.625;
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = 1.625d0
end function

Java

public static double code(double u1, double u2) {
	return 1.625;
}

Python

def code(u1, u2):
	return 1.625

Julia

function code(u1, u2)
	return 1.625
end

MATLAB

function tmp = code(u1, u2)
	tmp = 1.625;
end

Wolfram

code[u1_, u2_] := 1.625

Derivation

1. Initial program 99.4%

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

3. Applied egg-rr 18.1%

   \[\leadsto \color{blue}{1.125} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Taylor expanded in u2 around 0 18.1%

   \[\leadsto \color{blue}{1.625} \]

5. Final simplification 18.1%

   \[\leadsto 1.625 \]

Alternative 10: 19.4% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ 2.5 \end{array} \]

FPCore

(FPCore (u1 u2) :precision binary64 2.5)

C

double code(double u1, double u2) {
	return 2.5;
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = 2.5d0
end function

Java

public static double code(double u1, double u2) {
	return 2.5;
}

Python

def code(u1, u2):
	return 2.5

Julia

function code(u1, u2)
	return 2.5
end

MATLAB

function tmp = code(u1, u2)
	tmp = 2.5;
end

Wolfram

code[u1_, u2_] := 2.5

Derivation

1. Initial program 99.4%

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

3. Applied egg-rr 19.2%

   \[\leadsto \color{blue}{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Taylor expanded in u2 around 0 19.3%

   \[\leadsto \color{blue}{2.5} \]

5. Final simplification 19.3%

   \[\leadsto 2.5 \]

Alternative 11: 21.6% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ 4.5 \end{array} \]

FPCore

(FPCore (u1 u2) :precision binary64 4.5)

C

double code(double u1, double u2) {
	return 4.5;
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = 4.5d0
end function

Java

public static double code(double u1, double u2) {
	return 4.5;
}

Python

def code(u1, u2):
	return 4.5

Julia

function code(u1, u2)
	return 4.5
end

MATLAB

function tmp = code(u1, u2)
	tmp = 4.5;
end

Wolfram

code[u1_, u2_] := 4.5

Derivation

1. Initial program 99.4%

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

3. Applied egg-rr 21.8%

   \[\leadsto \color{blue}{4} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Taylor expanded in u2 around 0 21.8%

   \[\leadsto \color{blue}{4.5} \]

5. Final simplification 21.8%

   \[\leadsto 4.5 \]

Reproduce

herbie shell --seed 2023297 
(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))