normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 12.7s

Alternatives: 3

Speedup: 1.5×

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, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(Pi * N[(u2 + 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-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. 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. 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. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. rem-exp-log N/A

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

   4. unpow1/2 N/A

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

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({\left(e^{\log \left(-2 \cdot \log u1\right)}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{\frac{1}{36}}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   6. unpow1/2 N/A

      \[\leadsto \mathsf{fma}\left({\left(e^{\log \left(-2 \cdot \log u1\right)}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\frac{1}{36}}^{\frac{1}{2}}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   7. pow-prod-down N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(e^{\log \left(-2 \cdot \log u1\right)} \cdot \frac{1}{36}\right)}^{\frac{1}{2}}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   8. rem-exp-log N/A

      \[\leadsto \mathsf{fma}\left({\left(\color{blue}{\left(-2 \cdot \log u1\right)} \cdot \frac{1}{36}\right)}^{\frac{1}{2}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left({\color{blue}{\left(\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)\right)}}^{\frac{1}{2}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({\color{blue}{\left(\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)\right)}}^{\frac{1}{2}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   11. unpow1/2 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   12. lower-sqrt.f64 99.6

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{0.027777777777777776 \cdot \left(-2 \cdot \log u1\right)}}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{36} \cdot \color{blue}{\left(-2 \cdot \log u1\right)}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(\frac{1}{36} \cdot -2\right) \cdot \log u1}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1 \cdot \left(\frac{1}{36} \cdot -2\right)}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1 \cdot \left(\frac{1}{36} \cdot -2\right)}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   18. metadata-eval 99.6

       \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot \color{blue}{-0.05555555555555555}}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right) \]

6. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\log u1 \cdot -0.05555555555555555}}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right) \]
7. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot \frac{-1}{18}}, \cos \left(2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot u2\right)\right), \frac{1}{2}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot \frac{-1}{18}}, \cos \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2\right)}\right), \frac{1}{2}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot \frac{-1}{18}}, \cos \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)}, \frac{1}{2}\right) \]

   4. lift-cos.f64 99.6

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \color{blue}{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}, 0.5\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot \frac{-1}{18}}, \cos \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)}, \frac{1}{2}\right) \]

   6. count-2 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot \frac{-1}{18}}, \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)}, \frac{1}{2}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot \frac{-1}{18}}, \cos \left(\color{blue}{\mathsf{PI}\left(\right) \cdot u2} + \mathsf{PI}\left(\right) \cdot u2\right), \frac{1}{2}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot \frac{-1}{18}}, \cos \left(\mathsf{PI}\left(\right) \cdot u2 + \color{blue}{\mathsf{PI}\left(\right) \cdot u2}\right), \frac{1}{2}\right) \]

   9. distribute-lft-out N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot \frac{-1}{18}}, \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)}, \frac{1}{2}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot \frac{-1}{18}}, \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)}, \frac{1}{2}\right) \]

   11. lower-+.f64 99.6

       \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \cos \left(\pi \cdot \color{blue}{\left(u2 + u2\right)}\right), 0.5\right) \]

8. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \color{blue}{\cos \left(\pi \cdot \left(u2 + u2\right)\right)}, 0.5\right) \]
9. Add Preprocessing

Alternative 2: 99.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right), 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (sqrt (* (log u1) -0.05555555555555555))
  (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0)
  0.5))

C

double code(double u1, double u2) {
	return fma(sqrt((log(u1) * -0.05555555555555555)), fma(u2, (u2 * (-2.0 * (((double) M_PI) * ((double) M_PI)))), 1.0), 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(Float64(log(u1) * -0.05555555555555555)), fma(u2, Float64(u2 * Float64(-2.0 * Float64(pi * pi))), 1.0), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision] * N[(u2 * N[(u2 * N[(-2.0 * N[(Pi * Pi), $MachinePrecision]), $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. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. 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. 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. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. rem-exp-log N/A

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

   4. unpow1/2 N/A

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

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({\left(e^{\log \left(-2 \cdot \log u1\right)}\right)}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{\frac{1}{36}}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   6. unpow1/2 N/A

      \[\leadsto \mathsf{fma}\left({\left(e^{\log \left(-2 \cdot \log u1\right)}\right)}^{\frac{1}{2}} \cdot \color{blue}{{\frac{1}{36}}^{\frac{1}{2}}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   7. pow-prod-down N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(e^{\log \left(-2 \cdot \log u1\right)} \cdot \frac{1}{36}\right)}^{\frac{1}{2}}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   8. rem-exp-log N/A

      \[\leadsto \mathsf{fma}\left({\left(\color{blue}{\left(-2 \cdot \log u1\right)} \cdot \frac{1}{36}\right)}^{\frac{1}{2}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left({\color{blue}{\left(\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)\right)}}^{\frac{1}{2}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left({\color{blue}{\left(\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)\right)}}^{\frac{1}{2}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   11. unpow1/2 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   12. lower-sqrt.f64 99.6

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{0.027777777777777776 \cdot \left(-2 \cdot \log u1\right)}}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{36} \cdot \color{blue}{\left(-2 \cdot \log u1\right)}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(\frac{1}{36} \cdot -2\right) \cdot \log u1}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1 \cdot \left(\frac{1}{36} \cdot -2\right)}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1 \cdot \left(\frac{1}{36} \cdot -2\right)}}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right) \]

   18. metadata-eval 99.6

       \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot \color{blue}{-0.05555555555555555}}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right) \]

6. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\log u1 \cdot -0.05555555555555555}}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right) \]
7. Step-by-step derivation

   1. lift-approx 98.2

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \color{blue}{\mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\pi \cdot \pi\right), 1\right)}, 0.5\right) \]

8. Applied rewrites 98.2%

   \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)}, 0.5\right) \]
9. Add Preprocessing

Alternative 3: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, 1, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (sqrt (* (log u1) -0.05555555555555555)) 1.0 0.5))

C

double code(double u1, double u2) {
	return fma(sqrt((log(u1) * -0.05555555555555555)), 1.0, 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(Float64(log(u1) * -0.05555555555555555)), 1.0, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision] * 1.0 + 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. Step-by-step derivation

   1. lift-PI.f64 99.5

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

   2. rem-square-sqrt N/A

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

   3. sqrt-unprod N/A

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

   4. lift-PI.f64 N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-*r* N/A

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

   7. sqrt-prod N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 99.5

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, 1, 0.5\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(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))