normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 19.5s

Alternatives: 4

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double u1, double u2) {
	return (Math.sqrt((Math.log(u1) * -0.05555555555555555)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (math.sqrt((math.log(u1) * -0.05555555555555555)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

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

MATLAB

function tmp = code(u1, u2)
	tmp = (sqrt((log(u1) * -0.05555555555555555)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $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

4. Applied egg-rr 0

   \[\leadsto expr\]

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]
9. Add Preprocessing

Alternative 2: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot \pi\right)\\ 0.5 + \frac{\sqrt{\log u1 \cdot -0.05555555555555555} \cdot \left(1 + \left(\left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot -4\right)}{1 + 2 \cdot \left(u2 \cdot t\_0\right)} \end{array} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (let* ((t_0 (* u2 (* PI PI))))
   (+
    0.5
    (/
     (*
      (sqrt (* (log u1) -0.05555555555555555))
      (+ 1.0 (* (* (* u2 u2) (* t_0 t_0)) -4.0)))
     (+ 1.0 (* 2.0 (* u2 t_0)))))))

C

double code(double u1, double u2) {
	double t_0 = u2 * (((double) M_PI) * ((double) M_PI));
	return 0.5 + ((sqrt((log(u1) * -0.05555555555555555)) * (1.0 + (((u2 * u2) * (t_0 * t_0)) * -4.0))) / (1.0 + (2.0 * (u2 * t_0))));
}

Java

public static double code(double u1, double u2) {
	double t_0 = u2 * (Math.PI * Math.PI);
	return 0.5 + ((Math.sqrt((Math.log(u1) * -0.05555555555555555)) * (1.0 + (((u2 * u2) * (t_0 * t_0)) * -4.0))) / (1.0 + (2.0 * (u2 * t_0))));
}

Python

def code(u1, u2):
	t_0 = u2 * (math.pi * math.pi)
	return 0.5 + ((math.sqrt((math.log(u1) * -0.05555555555555555)) * (1.0 + (((u2 * u2) * (t_0 * t_0)) * -4.0))) / (1.0 + (2.0 * (u2 * t_0))))

Julia

function code(u1, u2)
	t_0 = Float64(u2 * Float64(pi * pi))
	return Float64(0.5 + Float64(Float64(sqrt(Float64(log(u1) * -0.05555555555555555)) * Float64(1.0 + Float64(Float64(Float64(u2 * u2) * Float64(t_0 * t_0)) * -4.0))) / Float64(1.0 + Float64(2.0 * Float64(u2 * t_0)))))
end

MATLAB

function tmp = code(u1, u2)
	t_0 = u2 * (pi * pi);
	tmp = 0.5 + ((sqrt((log(u1) * -0.05555555555555555)) * (1.0 + (((u2 * u2) * (t_0 * t_0)) * -4.0))) / (1.0 + (2.0 * (u2 * t_0))));
end

Wolfram

code[u1_, u2_] := Block[{t$95$0 = N[(u2 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, N[(0.5 + N[(N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(N[(u2 * u2), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(u2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

7. Taylor expanded in u2 around 0 0

   \[\leadsto expr\]

9. Simplified 0

   \[\leadsto expr\]

11. Applied egg-rr 0

    \[\leadsto expr\]
12. Add Preprocessing

Alternative 3: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 0.5 + \left(0.16666666666666666 + \left(u2 \cdot \left(u2 \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot -0.3333333333333333\right) \cdot \sqrt{-2 \cdot \log u1} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  0.5
  (*
   (+ 0.16666666666666666 (* (* u2 (* u2 (* PI PI))) -0.3333333333333333))
   (sqrt (* -2.0 (log u1))))))

C

double code(double u1, double u2) {
	return 0.5 + ((0.16666666666666666 + ((u2 * (u2 * (((double) M_PI) * ((double) M_PI)))) * -0.3333333333333333)) * sqrt((-2.0 * log(u1))));
}

Java

public static double code(double u1, double u2) {
	return 0.5 + ((0.16666666666666666 + ((u2 * (u2 * (Math.PI * Math.PI))) * -0.3333333333333333)) * Math.sqrt((-2.0 * Math.log(u1))));
}

Python

def code(u1, u2):
	return 0.5 + ((0.16666666666666666 + ((u2 * (u2 * (math.pi * math.pi))) * -0.3333333333333333)) * math.sqrt((-2.0 * math.log(u1))))

Julia

function code(u1, u2)
	return Float64(0.5 + Float64(Float64(0.16666666666666666 + Float64(Float64(u2 * Float64(u2 * Float64(pi * pi))) * -0.3333333333333333)) * sqrt(Float64(-2.0 * log(u1)))))
end

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + ((0.16666666666666666 + ((u2 * (u2 * (pi * pi))) * -0.3333333333333333)) * sqrt((-2.0 * log(u1))));
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[(0.16666666666666666 + N[(N[(u2 * N[(u2 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

7. Taylor expanded in u2 around 0 0

   \[\leadsto expr\]

9. Simplified 0

   \[\leadsto expr\]

11. Applied egg-rr 0

    \[\leadsto expr\]
12. Add Preprocessing

Alternative 4: 98.3% accurate, 1.5× 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%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in u2 around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]

11. Applied egg-rr 0

    \[\leadsto expr\]

13. Applied egg-rr 0

    \[\leadsto expr\]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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))