normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 13.2s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-pow.f64 N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(-2 \cdot \log u1\right)}^{0.25}, {\left(-2 \cdot \log u1\right)}^{0.25} \cdot \left(0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right), 0.5\right)} \]

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

7. Applied rewrites 99.5%

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

Alternative 2: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(0.16666666666666666 * N[Sqrt[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(2.0 * N[(u2 * Pi), $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-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 99.4

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.4

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

   7. lift-pow.f64 N/A

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

   8. unpow1/2 N/A

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

   9. lower-sqrt.f64 99.4

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

   10. lift-/.f64 N/A

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

   11. metadata-eval 99.4

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 4: 99.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(u2 * u2), $MachinePrecision] * N[(-0.3333333333333333 * N[(Pi * Pi), $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-pow.f64 N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(-2 \cdot \log u1\right)}^{0.25}, {\left(-2 \cdot \log u1\right)}^{0.25} \cdot \left(0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right), 0.5\right)} \]

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 98.1%

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

11. Taylor expanded in u2 around 0

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

13. Applied rewrites 98.9%

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

Alternative 5: 98.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{2 \cdot \left(-\log u1\right)}, \mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right), 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* 0.16666666666666666 (sqrt (* 2.0 (- (log u1)))))
  (fma (* PI PI) (* -2.0 (* u2 u2)) 1.0)
  0.5))

C

double code(double u1, double u2) {
	return fma((0.16666666666666666 * sqrt((2.0 * -log(u1)))), fma((((double) M_PI) * ((double) M_PI)), (-2.0 * (u2 * u2)), 1.0), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(0.16666666666666666 * sqrt(Float64(2.0 * Float64(-log(u1))))), fma(Float64(pi * pi), Float64(-2.0 * Float64(u2 * u2)), 1.0), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(0.16666666666666666 * N[Sqrt[N[(2.0 * (-N[Log[u1], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(-2.0 * N[(u2 * u2), $MachinePrecision]), $MachinePrecision] + 1.0), $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 u1 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. log-rec N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-sqrt.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. rem-square-sqrt N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-PI.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. rem-square-sqrt 98.8

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

8. Applied rewrites 98.8%

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\pi \cdot \pi, \left(u2 \cdot u2\right) \cdot -2, 1\right)} + 0.5 \]
9. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 98.8

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

10. Applied rewrites 98.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{-\log u1 \cdot 2}, \mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right), 0.5\right)} \]

11. Final simplification 98.7%

    \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{2 \cdot \left(-\log u1\right)}, \mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right), 0.5\right) \]
12. Add Preprocessing

Alternative 6: 98.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-pow.f64 N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(-2 \cdot \log u1\right)}^{0.25}, {\left(-2 \cdot \log u1\right)}^{0.25} \cdot \left(0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right), 0.5\right)} \]

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 98.1%

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

Alternative 7: 98.4% accurate, 2.5× speedup

Math

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

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(sqrt(2.0), Float64(0.16666666666666666 * sqrt(Float64(-log(u1)))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(0.16666666666666666 * N[Sqrt[(-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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-pow.f64 N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(-2 \cdot \log u1\right)}^{0.25}, {\left(-2 \cdot \log u1\right)}^{0.25} \cdot \left(0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right), 0.5\right)} \]

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

10. Applied rewrites 99.5%

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

11. Taylor expanded in u2 around 0

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

13. Applied rewrites 98.0%

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

Alternative 8: 98.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 0.0

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

5. Applied rewrites 0.0%

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

7. Applied rewrites 97.9%

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

Reproduce

herbie shell --seed 2024219 
(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))