normal distribution

Percentage Accurate: 99.4% → 99.4%

Time: 4.3s

Alternatives: 5

Speedup: 1.4×

Specification

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

Math

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

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

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

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: 99.4% accurate, 1.4× speedup

Math

\[\mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666, \cos \left(u2 \cdot 6.283185307179586\right), 0.5\right) \]

FPCore

(FPCore (u1 u2)
  :precision binary64
  (fma
 (* (sqrt (* (log u1) -2.0)) 0.16666666666666666)
 (cos (* u2 6.283185307179586))
 0.5))

C

double code(double u1, double u2) {
	return fma((sqrt((log(u1) * -2.0)) * 0.16666666666666666), cos((u2 * 6.283185307179586)), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(sqrt(Float64(log(u1) * -2.0)) * 0.16666666666666666), cos(Float64(u2 * 6.283185307179586)), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Cos[N[(u2 * 6.283185307179586), $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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   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

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

   5. *-commutative N/A

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

   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

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

   8. unpow1/2 N/A

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

   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

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.4%

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

   13. lift-/.f64 N/A

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

   14. metadata-eval 99.4%

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.4%

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

   18. lift-*.f64 N/A

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

   19. count-2-rev N/A

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

   20. lower-+.f64 99.4%

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

3. Applied rewrites 99.4%

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

4. Evaluated real constant 99.4%

   \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666, \cos \left(u2 \cdot \color{blue}{6.283185307179586}\right), 0.5\right) \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.4× speedup

Math

\[\mathsf{fma}\left(\cos \left(-6.283185307179586 \cdot u2\right) \cdot \sqrt{-2 \cdot \log u1}, 0.16666666666666666, 0.5\right) \]

FPCore

(FPCore (u1 u2)
  :precision binary64
  (fma
 (* (cos (* -6.283185307179586 u2)) (sqrt (* -2.0 (log u1))))
 0.16666666666666666
 0.5))

C

double code(double u1, double u2) {
	return fma((cos((-6.283185307179586 * u2)) * sqrt((-2.0 * log(u1)))), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(cos(Float64(-6.283185307179586 * u2)) * sqrt(Float64(-2.0 * log(u1)))), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(-6.283185307179586 * u2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   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

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

   5. *-commutative N/A

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

   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

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

   8. unpow1/2 N/A

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

   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

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.4%

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

   13. lift-/.f64 N/A

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

   14. metadata-eval 99.4%

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.4%

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

   18. lift-*.f64 N/A

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

   19. count-2-rev N/A

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

   20. lower-+.f64 99.4%

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

3. Applied rewrites 99.4%

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

4. Evaluated real constant 99.4%

   \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666, \cos \left(u2 \cdot \color{blue}{6.283185307179586}\right), 0.5\right) \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{\log u1 \cdot -2} \cdot \frac{1}{6}\right) \cdot \cos \left(u2 \cdot \frac{884279719003555}{140737488355328}\right) + \frac{1}{2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{\log u1 \cdot -2} \cdot \frac{1}{6}\right)} \cdot \cos \left(u2 \cdot \frac{884279719003555}{140737488355328}\right) + \frac{1}{2} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\sqrt{\log u1 \cdot -2} \cdot \left(\frac{1}{6} \cdot \cos \left(u2 \cdot \frac{884279719003555}{140737488355328}\right)\right)} + \frac{1}{2} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\log u1 \cdot -2}} \cdot \left(\frac{1}{6} \cdot \cos \left(u2 \cdot \frac{884279719003555}{140737488355328}\right)\right) + \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \log u1}} \cdot \left(\frac{1}{6} \cdot \cos \left(u2 \cdot \frac{884279719003555}{140737488355328}\right)\right) + \frac{1}{2} \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \log u1}} \cdot \left(\frac{1}{6} \cdot \cos \left(u2 \cdot \frac{884279719003555}{140737488355328}\right)\right) + \frac{1}{2} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \cos \left(u2 \cdot \frac{884279719003555}{140737488355328}\right)\right) \cdot \sqrt{-2 \cdot \log u1}} + \frac{1}{2} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\left(\cos \left(u2 \cdot \frac{884279719003555}{140737488355328}\right) \cdot \frac{1}{6}\right)} \cdot \sqrt{-2 \cdot \log u1} + \frac{1}{2} \]

   9. associate-*r* N/A

      \[\leadsto \color{blue}{\cos \left(u2 \cdot \frac{884279719003555}{140737488355328}\right) \cdot \left(\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}\right)} + \frac{1}{2} \]

   10. *-commutative N/A

       \[\leadsto \cos \left(u2 \cdot \frac{884279719003555}{140737488355328}\right) \cdot \color{blue}{\left(\sqrt{-2 \cdot \log u1} \cdot \frac{1}{6}\right)} + \frac{1}{2} \]

   11. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\cos \left(u2 \cdot \frac{884279719003555}{140737488355328}\right) \cdot \sqrt{-2 \cdot \log u1}\right) \cdot \frac{1}{6}} + \frac{1}{2} \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(u2 \cdot \frac{884279719003555}{140737488355328}\right) \cdot \sqrt{-2 \cdot \log u1}, \frac{1}{6}, \frac{1}{2}\right)} \]

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-6.283185307179586 \cdot u2\right) \cdot \sqrt{-2 \cdot \log u1}, 0.16666666666666666, 0.5\right)} \]
7. Add Preprocessing

Alternative 3: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{-2 \cdot \log u1}\\ \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -0.3333333333333333, 9.869604401089358 \cdot t\_0, t\_0 \cdot 0.16666666666666666\right) - -0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
  :precision binary64
  (let* ((t_0 (sqrt (* -2.0 (log u1)))))
  (-
   (fma
    (* (* u2 u2) -0.3333333333333333)
    (* 9.869604401089358 t_0)
    (* t_0 0.16666666666666666))
   -0.5)))

C

double code(double u1, double u2) {
	double t_0 = sqrt((-2.0 * log(u1)));
	return fma(((u2 * u2) * -0.3333333333333333), (9.869604401089358 * t_0), (t_0 * 0.16666666666666666)) - -0.5;
}

Julia

function code(u1, u2)
	t_0 = sqrt(Float64(-2.0 * log(u1)))
	return Float64(fma(Float64(Float64(u2 * u2) * -0.3333333333333333), Float64(9.869604401089358 * t_0), Float64(t_0 * 0.16666666666666666)) - -0.5)
end

Wolfram

code[u1_, u2_] := Block[{t$95$0 = N[Sqrt[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(u2 * u2), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(9.869604401089358 * t$95$0), $MachinePrecision] + N[(t$95$0 * 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. Taylor expanded in u2 around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-log.f64 98.8%

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.8%

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

7. Evaluated real constant 98.8%

   \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -0.3333333333333333, 9.869604401089358 \cdot \sqrt{\color{blue}{-2 \cdot \log u1}}, \sqrt{-2 \cdot \log u1} \cdot 0.16666666666666666\right) - -0.5 \]
8. Add Preprocessing

Alternative 4: 98.8% accurate, 2.8× speedup

Math

\[\sqrt{-2 \cdot \log u1} \cdot \mathsf{fma}\left(9.869604401089358, -0.3333333333333333 \cdot \left(u2 \cdot u2\right), 0.16666666666666666\right) - -0.5 \]

FPCore

(FPCore (u1 u2)
  :precision binary64
  (-
 (*
  (sqrt (* -2.0 (log u1)))
  (fma
   9.869604401089358
   (* -0.3333333333333333 (* u2 u2))
   0.16666666666666666))
 -0.5))

C

double code(double u1, double u2) {
	return (sqrt((-2.0 * log(u1))) * fma(9.869604401089358, (-0.3333333333333333 * (u2 * u2)), 0.16666666666666666)) - -0.5;
}

Julia

function code(u1, u2)
	return Float64(Float64(sqrt(Float64(-2.0 * log(u1))) * fma(9.869604401089358, Float64(-0.3333333333333333 * Float64(u2 * u2)), 0.16666666666666666)) - -0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(9.869604401089358 * N[(-0.3333333333333333 * N[(u2 * u2), $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. Taylor expanded in u2 around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-log.f64 98.8%

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

4. Applied rewrites 98.8%

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

6. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -0.3333333333333333, \left(\pi \cdot \pi\right) \cdot \sqrt{-2 \cdot \log u1}, \sqrt{-2 \cdot \log u1} \cdot 0.16666666666666666\right) - -0.5} \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-out N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 98.8%

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 98.8%

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

8. Applied rewrites 98.8%

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

9. Evaluated real constant 98.8%

   \[\leadsto \sqrt{-2 \cdot \log u1} \cdot \mathsf{fma}\left(9.869604401089358, \color{blue}{-0.3333333333333333} \cdot \left(u2 \cdot u2\right), 0.16666666666666666\right) - -0.5 \]
10. Add Preprocessing

Alternative 5: 98.2% accurate, 4.6× speedup

Math

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

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. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}} \]
3. Step-by-step derivation

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 98.2%

      \[\leadsto 0.5 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1} \]

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{0.5 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 98.2%

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

6. Applied rewrites 98.2%

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

Reproduce

herbie shell --seed 2025212 
(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))