normal distribution

Percentage Accurate: 6.8% → 9.2%

Time: 8.9s

Alternatives: 6

Speedup: 5.1×

• Could not converge on a ground truth

  1. u1 = 1.7533746046869366e+195
  2. u2 = -1.7030234586554808e+258

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: 6.8% 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: 9.2% accurate, 0.8× speedup

Math

\[\left(\sqrt{\left|\log \left(\left|u1\right|\right)\right|} \cdot \cos \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(u2 + u2\right)\right)\right)\right) \cdot 0.23570226039551584 + 0.5 \]

FPCore

(FPCore (u1 u2)
  :precision binary64
  (+
 (*
  (*
   (sqrt (fabs (log (fabs u1))))
   (cos (* (cbrt (* PI PI)) (* (cbrt PI) (+ u2 u2)))))
  0.23570226039551584)
 0.5))

C

double code(double u1, double u2) {
	return ((sqrt(fabs(log(fabs(u1)))) * cos((cbrt((((double) M_PI) * ((double) M_PI))) * (cbrt(((double) M_PI)) * (u2 + u2))))) * 0.23570226039551584) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return ((Math.sqrt(Math.abs(Math.log(Math.abs(u1)))) * Math.cos((Math.cbrt((Math.PI * Math.PI)) * (Math.cbrt(Math.PI) * (u2 + u2))))) * 0.23570226039551584) + 0.5;
}

Julia

function code(u1, u2)
	return Float64(Float64(Float64(sqrt(abs(log(abs(u1)))) * cos(Float64(cbrt(Float64(pi * pi)) * Float64(cbrt(pi) * Float64(u2 + u2))))) * 0.23570226039551584) + 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[Sqrt[N[Abs[N[Log[N[Abs[u1], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[Power[N[(Pi * Pi), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[(u2 + u2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.23570226039551584), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. sqrt-prod N/A

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

   6. metadata-eval N/A

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

   7. sqrt-unprod N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-fabs.f64 7.6%

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

3. Applied rewrites 7.6%

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

5. Applied rewrites 7.6%

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

6. Evaluated real constant 7.6%

   \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\left(u2 + u2\right) \cdot \pi\right)\right) \cdot \color{blue}{0.23570226039551584} + 0.5 \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \color{blue}{\left(\left(u2 + u2\right) \cdot \pi\right)}\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   2. lift-+.f64 N/A

      \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\color{blue}{\left(u2 + u2\right)} \cdot \pi\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   3. count-2 N/A

      \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\color{blue}{\left(2 \cdot u2\right)} \cdot \pi\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\color{blue}{\left(u2 \cdot 2\right)} \cdot \pi\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \color{blue}{\left(\pi \cdot \left(u2 \cdot 2\right)\right)}\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   6. lift-PI.f64 N/A

      \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   7. add-cube-cbrt N/A

      \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \left(u2 \cdot 2\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   8. associate-*l* N/A

      \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)\right)}\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)\right)}\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   10. lift-PI.f64 N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   11. lift-PI.f64 N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   12. cbrt-unprod N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\color{blue}{\sqrt[3]{\pi \cdot \pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   13. lift-PI.f64 N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)} \cdot \pi} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   14. lift-PI.f64 N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   15. lower-cbrt.f64 N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   16. lift-PI.f64 N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\sqrt[3]{\color{blue}{\pi} \cdot \mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   17. lift-PI.f64 N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\sqrt[3]{\pi \cdot \color{blue}{\pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   18. lower-*.f64 N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\sqrt[3]{\color{blue}{\pi \cdot \pi}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   19. lower-*.f64 N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\sqrt[3]{\pi \cdot \pi} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(u2 \cdot 2\right)\right)}\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   20. lift-PI.f64 N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \left(u2 \cdot 2\right)\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   21. lower-cbrt.f64 N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot \left(u2 \cdot 2\right)\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   22. *-commutative N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(2 \cdot u2\right)}\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   23. count-2 N/A

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(u2 + u2\right)}\right)\right)\right) \cdot \frac{4246034448350515}{18014398509481984} + \frac{1}{2} \]

   24. lift-+.f64 7.6%

       \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(u2 + u2\right)}\right)\right)\right) \cdot 0.23570226039551584 + 0.5 \]

8. Applied rewrites 7.6%

   \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \color{blue}{\left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(u2 + u2\right)\right)\right)}\right) \cdot 0.23570226039551584 + 0.5 \]
9. Add Preprocessing

Alternative 2: 9.2% accurate, 1.4× speedup

Math

\[\mathsf{fma}\left(\cos \left(\left(\pi + \pi\right) \cdot u2\right) \cdot 0.23570226039551584, \sqrt{\left|\log \left(\left|u1\right|\right)\right|}, 0.5\right) \]

FPCore

(FPCore (u1 u2)
  :precision binary64
  (fma
 (* (cos (* (+ PI PI) u2)) 0.23570226039551584)
 (sqrt (fabs (log (fabs u1))))
 0.5))

C

double code(double u1, double u2) {
	return fma((cos(((((double) M_PI) + ((double) M_PI)) * u2)) * 0.23570226039551584), sqrt(fabs(log(fabs(u1)))), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(cos(Float64(Float64(pi + pi) * u2)) * 0.23570226039551584), sqrt(abs(log(abs(u1)))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(N[(Pi + Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision] * 0.23570226039551584), $MachinePrecision] * N[Sqrt[N[Abs[N[Log[N[Abs[u1], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. sqrt-prod N/A

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

   6. metadata-eval N/A

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

   7. sqrt-unprod N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-fabs.f64 7.6%

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

3. Applied rewrites 7.6%

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

5. Applied rewrites 7.6%

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

6. Evaluated real constant 7.6%

   \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\left(u2 + u2\right) \cdot \pi\right)\right) \cdot \color{blue}{0.23570226039551584} + 0.5 \]

8. Applied rewrites 7.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(\pi + \pi\right) \cdot u2\right) \cdot 0.23570226039551584, \sqrt{\left|\log u1\right|}, 0.5\right)} \]
9. Add Preprocessing

Alternative 3: 9.1% accurate, 1.4× speedup

Math

\[\mathsf{fma}\left(0.23570226039551584 \cdot \sqrt{\left|\log \left(\left|u1\right|\right)\right|}, \cos \left(\left(\pi + \pi\right) \cdot u2\right), 0.5\right) \]

FPCore

(FPCore (u1 u2)
  :precision binary64
  (fma
 (* 0.23570226039551584 (sqrt (fabs (log (fabs u1)))))
 (cos (* (+ PI PI) u2))
 0.5))

C

double code(double u1, double u2) {
	return fma((0.23570226039551584 * sqrt(fabs(log(fabs(u1))))), cos(((((double) M_PI) + ((double) M_PI)) * u2)), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(0.23570226039551584 * sqrt(abs(log(abs(u1))))), cos(Float64(Float64(pi + pi) * u2)), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(0.23570226039551584 * N[Sqrt[N[Abs[N[Log[N[Abs[u1], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(Pi + Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. sqrt-prod N/A

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

   6. metadata-eval N/A

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

   7. sqrt-unprod N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-fabs.f64 7.6%

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

3. Applied rewrites 7.6%

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

5. Applied rewrites 7.6%

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

6. Evaluated real constant 7.6%

   \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\left(u2 + u2\right) \cdot \pi\right)\right) \cdot \color{blue}{0.23570226039551584} + 0.5 \]

8. Applied rewrites 7.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.23570226039551584 \cdot \sqrt{\left|\log u1\right|}, \cos \left(\left(\pi + \pi\right) \cdot u2\right), 0.5\right)} \]
9. Add Preprocessing

Alternative 4: 9.1% accurate, 4.8× speedup

Math

\[0.23570226039551584 \cdot \sqrt{\left|\log \left(\left|u1\right|\right)\right|} + 0.5 \]

FPCore

(FPCore (u1 u2)
  :precision binary64
  (+ (* 0.23570226039551584 (sqrt (fabs (log (fabs u1))))) 0.5))

C

double code(double u1, double u2) {
	return (0.23570226039551584 * sqrt(fabs(log(fabs(u1))))) + 0.5;
}

Fortran

real(8) function code(u1, u2)
use fmin_fmax_functions
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = (0.23570226039551584d0 * sqrt(abs(log(abs(u1))))) + 0.5d0
end function

Java

public static double code(double u1, double u2) {
	return (0.23570226039551584 * Math.sqrt(Math.abs(Math.log(Math.abs(u1))))) + 0.5;
}

Python

def code(u1, u2):
	return (0.23570226039551584 * math.sqrt(math.fabs(math.log(math.fabs(u1))))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(0.23570226039551584 * sqrt(abs(log(abs(u1))))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (0.23570226039551584 * sqrt(abs(log(abs(u1))))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(0.23570226039551584 * N[Sqrt[N[Abs[N[Log[N[Abs[u1], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. sqrt-prod N/A

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

   6. metadata-eval N/A

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

   7. sqrt-unprod N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-fabs.f64 7.6%

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

3. Applied rewrites 7.6%

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

5. Applied rewrites 7.6%

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

6. Evaluated real constant 7.6%

   \[\leadsto \left(\sqrt{\left|\log u1\right|} \cdot \cos \left(\left(u2 + u2\right) \cdot \pi\right)\right) \cdot \color{blue}{0.23570226039551584} + 0.5 \]

7. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{4246034448350515}{18014398509481984} \cdot \sqrt{\left|\log u1\right|}} + 0.5 \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{4246034448350515}{18014398509481984} \cdot \color{blue}{\sqrt{\left|\log u1\right|}} + \frac{1}{2} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{4246034448350515}{18014398509481984} \cdot \sqrt{\left|\log u1\right|} + \frac{1}{2} \]

   3. lower-fabs.f64 N/A

      \[\leadsto \frac{4246034448350515}{18014398509481984} \cdot \sqrt{\left|\log u1\right|} + \frac{1}{2} \]

   4. lower-log.f64 7.5%

      \[\leadsto 0.23570226039551584 \cdot \sqrt{\left|\log u1\right|} + 0.5 \]

9. Applied rewrites 7.5%

   \[\leadsto \color{blue}{0.23570226039551584 \cdot \sqrt{\left|\log u1\right|}} + 0.5 \]
10. Add Preprocessing

Alternative 5: 9.1% accurate, 4.8× speedup

Math

\[0.5 + \sqrt{\left|-0.05555555555555555 \cdot \log \left(\left|u1\right|\right)\right|} \]

FPCore

(FPCore (u1 u2)
  :precision binary64
  (+ 0.5 (sqrt (fabs (* -0.05555555555555555 (log (fabs u1)))))))

C

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

Fortran

real(8) function code(u1, u2)
use fmin_fmax_functions
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = 0.5d0 + sqrt(abs(((-0.05555555555555555d0) * log(abs(u1)))))
end function

Java

public static double code(double u1, double u2) {
	return 0.5 + Math.sqrt(Math.abs((-0.05555555555555555 * Math.log(Math.abs(u1)))));
}

Python

def code(u1, u2):
	return 0.5 + math.sqrt(math.fabs((-0.05555555555555555 * math.log(math.fabs(u1)))))

Julia

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

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + sqrt(abs((-0.05555555555555555 * log(abs(u1)))));
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[Sqrt[N[Abs[N[(-0.05555555555555555 * N[Log[N[Abs[u1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

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. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 6.8%

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

4. Applied rewrites 6.8%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-pow.f64 N/A

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

   11. pow1/2 N/A

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

   12. lift-*.f64 N/A

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

   13. sqrt-prod N/A

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

   14. metadata-eval N/A

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

   15. lift-fabs.f64 N/A

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

   16. sqrt-unprod N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lift-sqrt.f64 N/A

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

6. Applied rewrites 6.8%

   \[\leadsto 0.5 + \sqrt{0.027777777777777776 \cdot \left(\log u1 \cdot -2\right)} \]
7. Step-by-step derivation

   1. rem-square-sqrt N/A

      \[\leadsto \frac{1}{2} + \sqrt{\sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)} \cdot \sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} + \sqrt{\sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)} \cdot \sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{2} + \sqrt{\sqrt{\left(\log u1 \cdot -2\right) \cdot \frac{1}{36}} \cdot \sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)}} \]

   4. sqrt-unprod N/A

      \[\leadsto \frac{1}{2} + \sqrt{\left(\sqrt{\log u1 \cdot -2} \cdot \sqrt{\frac{1}{36}}\right) \cdot \sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)}} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} + \sqrt{\left(\sqrt{\log u1 \cdot -2} \cdot \sqrt{\frac{1}{36}}\right) \cdot \sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{2} + \sqrt{\left(\sqrt{\log u1 \cdot -2} \cdot \frac{1}{6}\right) \cdot \sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} + \sqrt{\left(\sqrt{\log u1 \cdot -2} \cdot \frac{1}{6}\right) \cdot \sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} + \sqrt{\left(\sqrt{\log u1 \cdot -2} \cdot \frac{1}{6}\right) \cdot \sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{2} + \sqrt{\left(\sqrt{\log u1 \cdot -2} \cdot \frac{1}{6}\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot \frac{1}{36}}} \]

   10. sqrt-unprod N/A

       \[\leadsto \frac{1}{2} + \sqrt{\left(\sqrt{\log u1 \cdot -2} \cdot \frac{1}{6}\right) \cdot \left(\sqrt{\log u1 \cdot -2} \cdot \sqrt{\frac{1}{36}}\right)} \]

   11. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{2} + \sqrt{\left(\sqrt{\log u1 \cdot -2} \cdot \frac{1}{6}\right) \cdot \left(\sqrt{\log u1 \cdot -2} \cdot \sqrt{\frac{1}{36}}\right)} \]

   12. metadata-eval N/A

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

   13. lift-*.f64 N/A

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

   14. sqr-abs-rev N/A

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

   15. mul-fabs N/A

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

8. Applied rewrites 7.6%

   \[\leadsto 0.5 + \sqrt{\left|-0.05555555555555555 \cdot \log u1\right|} \]
9. Add Preprocessing

Alternative 6: 7.6% accurate, 5.1× speedup

Math

\[\sqrt{-0.05555555555555555 \cdot \log \left(\left|u1\right|\right)} - -0.5 \]

FPCore

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

C

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

Fortran

real(8) function code(u1, u2)
use fmin_fmax_functions
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = sqrt(((-0.05555555555555555d0) * log(abs(u1)))) - (-0.5d0)
end function

Java

public static double code(double u1, double u2) {
	return Math.sqrt((-0.05555555555555555 * Math.log(Math.abs(u1)))) - -0.5;
}

Python

def code(u1, u2):
	return math.sqrt((-0.05555555555555555 * math.log(math.fabs(u1)))) - -0.5

Julia

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

MATLAB

function tmp = code(u1, u2)
	tmp = sqrt((-0.05555555555555555 * log(abs(u1)))) - -0.5;
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(-0.05555555555555555 * N[Log[N[Abs[u1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -0.5), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

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. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 6.8%

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

4. Applied rewrites 6.8%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-pow.f64 N/A

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

   11. pow1/2 N/A

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

   12. lift-*.f64 N/A

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

   13. sqrt-prod N/A

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

   14. metadata-eval N/A

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

   15. lift-fabs.f64 N/A

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

   16. sqrt-unprod N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lift-sqrt.f64 N/A

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

6. Applied rewrites 6.8%

   \[\leadsto 0.5 + \sqrt{0.027777777777777776 \cdot \left(\log u1 \cdot -2\right)} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{1}{2} + \color{blue}{\sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)}} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)} + \color{blue}{\frac{1}{2}} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)} + \frac{1}{2} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{36} \cdot \left(\log u1 \cdot -2\right)} + \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\left(\log u1 \cdot -2\right) \cdot \frac{1}{36}} + \frac{1}{2} \]

   6. sqrt-unprod N/A

      \[\leadsto \sqrt{\log u1 \cdot -2} \cdot \sqrt{\frac{1}{36}} + \frac{1}{2} \]

   7. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{\log u1 \cdot -2} \cdot \sqrt{\frac{1}{36}} + \frac{1}{2} \]

   8. metadata-eval N/A

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

   9. lift-*.f64 N/A

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

   10. add-flip N/A

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

   11. lower--.f64 N/A

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

8. Applied rewrites 6.8%

   \[\leadsto \sqrt{-0.05555555555555555 \cdot \log u1} - \color{blue}{-0.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025313 -o setup:search
(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))