normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 4.0s

Alternatives: 8

Speedup: 1.4×

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, 1.4× 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, 1.4× 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, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.4%

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

Alternative 4: 98.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[(-0.3333333333333333 * N[(u2 * u2), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[2.0], $MachinePrecision] * 0.16666666666666666), $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. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-sqrt.f64 98.4

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-PI.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-*.f64 99.3

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

10. Applied rewrites 99.3%

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

Alternative 5: 98.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * 0.16666666666666666), $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. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-sqrt.f64 98.4

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

7. Applied rewrites 98.4%

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

Alternative 6: 98.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right)} \]
4. Step-by-step derivation

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

      \[\leadsto \left(\sqrt{\log u1} \cdot \sqrt{-2}\right) \cdot \frac{1}{6} + \frac{1}{2} \]

   4. sqrt-unprod N/A

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

   5. *-commutative N/A

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

   6. unpow1/2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-log.f64 N/A

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

   13. metadata-eval 98.3

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

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. sqrt-prod N/A

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

   5. metadata-eval N/A

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

   6. sqrt-unprod N/A

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

   7. associate-*r* N/A

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

   8. sqrt-unprod N/A

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

   9. *-commutative N/A

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

   10. mul-1-neg N/A

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

   11. neg-log N/A

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

   12. lower-*.f64 N/A

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

   13. neg-log N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lift-log.f64 N/A

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

   16. lift-neg.f64 N/A

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

   17. lift-sqrt.f64 98.3

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

7. Applied rewrites 98.3%

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

Alternative 7: 98.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right)} \]
4. Step-by-step derivation

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

      \[\leadsto \left(\sqrt{\log u1} \cdot \sqrt{-2}\right) \cdot \frac{1}{6} + \frac{1}{2} \]

   4. sqrt-unprod N/A

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

   5. *-commutative N/A

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

   6. unpow1/2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-log.f64 N/A

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

   13. metadata-eval 98.3

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

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right)} \]
6. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. fp-cancel-sub-sign-inv N/A

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

   9. metadata-eval N/A

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

   10. sqrt-prod N/A

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

   11. metadata-eval N/A

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

   12. sqrt-unprod N/A

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

   13. associate-*r* N/A

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

   14. sqrt-unprod N/A

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

   15. *-commutative N/A

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

   16. mul-1-neg N/A

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

   17. neg-log N/A

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

   18. metadata-eval N/A

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

   19. sqrt-unprod N/A

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

7. Applied rewrites 98.3%

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

Alternative 8: 98.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right)} \]
4. Step-by-step derivation

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

      \[\leadsto \left(\sqrt{\log u1} \cdot \sqrt{-2}\right) \cdot \frac{1}{6} + \frac{1}{2} \]

   4. sqrt-unprod N/A

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

   5. *-commutative N/A

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

   6. unpow1/2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-log.f64 N/A

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

   13. metadata-eval 98.3

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

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right)} \]
6. 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))