normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 8.7s

Alternatives: 4

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[(0.16666666666666666 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(Pi * u2), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $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 99.6%

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

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

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

Alternative 3: 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 4: 98.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\log u1 \cdot -2\right) \cdot 0.5\\ \left(\frac{1}{6} \cdot \left(\cosh t\_0 + \sinh t\_0\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (let* ((t_0 (* (log (* (log u1) -2.0)) 0.5)))
   (+
    (* (* (/ 1.0 6.0) (+ (cosh t_0) (sinh t_0))) (cos (* (* 2.0 PI) u2)))
    0.5)))

C

double code(double u1, double u2) {
	double t_0 = log((log(u1) * -2.0)) * 0.5;
	return (((1.0 / 6.0) * (cosh(t_0) + sinh(t_0))) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	double t_0 = Math.log((Math.log(u1) * -2.0)) * 0.5;
	return (((1.0 / 6.0) * (Math.cosh(t_0) + Math.sinh(t_0))) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	t_0 = math.log((math.log(u1) * -2.0)) * 0.5
	return (((1.0 / 6.0) * (math.cosh(t_0) + math.sinh(t_0))) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	t_0 = Float64(log(Float64(log(u1) * -2.0)) * 0.5)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * Float64(cosh(t_0) + sinh(t_0))) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	t_0 = log((log(u1) * -2.0)) * 0.5;
	tmp = (((1.0 / 6.0) * (cosh(t_0) + sinh(t_0))) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := Block[{t$95$0 = N[(N[Log[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[(N[Cosh[t$95$0], $MachinePrecision] + N[Sinh[t$95$0], $MachinePrecision]), $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-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. sinh-+-cosh-rev N/A

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

   6. lower-+.f64 N/A

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

   7. lower-cosh.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-log.f64 N/A

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

   13. lower-sinh.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-log.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-log.f64 99.0

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

4. Applied rewrites 99.0%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\cosh \left(\log \left(\log u1 \cdot -2\right) \cdot 0.5\right) + \sinh \left(\log \left(\log u1 \cdot -2\right) \cdot 0.5\right)\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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))