normal distribution

Percentage Accurate: 99.4% → 99.5%

Time: 7.7s

Alternatives: 8

Speedup: 1.3×

Specification

\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.5%

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

Alternative 3: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in u1 around 0

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

6. Applied rewrites 99.4%

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

8. Applied rewrites 99.4%

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

Alternative 4: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-2 \cdot \log u1}\\ \frac{\mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot t\_0, t\_0\right)}{6} + 0.5 \end{array} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (let* ((t_0 (sqrt (* -2.0 (log u1)))))
   (+ (/ (fma (* -2.0 (* u2 u2)) (* (* PI PI) t_0) t_0) 6.0) 0.5)))

C

double code(double u1, double u2) {
	double t_0 = sqrt((-2.0 * log(u1)));
	return (fma((-2.0 * (u2 * u2)), ((((double) M_PI) * ((double) M_PI)) * t_0), t_0) / 6.0) + 0.5;
}

Julia

function code(u1, u2)
	t_0 = sqrt(Float64(-2.0 * log(u1)))
	return Float64(Float64(fma(Float64(-2.0 * Float64(u2 * u2)), Float64(Float64(pi * pi) * t_0), t_0) / 6.0) + 0.5)
end

Wolfram

code[u1_, u2_] := Block[{t$95$0 = N[Sqrt[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(-2.0 * N[(u2 * u2), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$0), $MachinePrecision] / 6.0), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in u2 around 0

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

6. Applied rewrites 98.8%

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

Alternative 5: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in u2 around 0

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

4. Applied rewrites 98.6%

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

6. Applied rewrites 98.8%

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

Alternative 6: 98.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, \color{blue}{\frac{1}{6} + \frac{-1}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \frac{1}{2}\right) \]

8. Applied rewrites 98.6%

   \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, \color{blue}{\mathsf{fma}\left(-0.3333333333333333, \left(\pi \cdot \pi\right) \cdot \left(u2 \cdot u2\right), 0.16666666666666666\right)}, 0.5\right) \]
9. Add Preprocessing

Alternative 7: 98.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in u2 around 0

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

6. Applied rewrites 98.2%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \frac{\sqrt{-2 \cdot \log u1}}{6}, 0.5\right) \]
7. Add Preprocessing

Alternative 8: 98.0% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Applied rewrites 98.0%

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

Reproduce

herbie shell --seed 2025139 
(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))