normal distribution

Percentage Accurate: 99.4% → 99.7%

Time: 13.8s

Alternatives: 5

Speedup: 1.0×

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: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\mathsf{log1p}\left({u1}^{-0.05555555555555555} + -1\right)}, \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right), 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (sqrt (log1p (+ (pow u1 -0.05555555555555555) -1.0)))
  (cos (* u2 (* 2.0 PI)))
  0.5))

C

double code(double u1, double u2) {
	return fma(sqrt(log1p((pow(u1, -0.05555555555555555) + -1.0))), cos((u2 * (2.0 * ((double) M_PI)))), 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(log1p(Float64((u1 ^ -0.05555555555555555) + -1.0))), cos(Float64(u2 * Float64(2.0 * pi))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[Log[1 + N[(N[Power[u1, -0.05555555555555555], $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(u2 * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. pow1/2 99.4%

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

   2. add-log-exp 99.4%

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

   3. metadata-eval 99.4%

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

   4. *-un-lft-identity 99.4%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   5. metadata-eval 99.4%

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

   6. log-prod 99.4%

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

   7. metadata-eval 99.4%

      \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   8. add-log-exp 99.4%

      \[\leadsto \left(0 + \color{blue}{\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   9. metadata-eval 99.4%

      \[\leadsto \left(0 + \color{blue}{0.16666666666666666} \cdot \sqrt{-2 \cdot \log u1}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\left(0 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

   1. +-lft-identity 99.4%

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

   2. *-commutative 99.4%

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

6. Simplified 99.4%

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

8. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} - 1} \]
9. Step-by-step derivation

   1. sub-neg 99.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} + \left(-1\right)} \]

   2. metadata-eval 99.1%

      \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} + \color{blue}{-1} \]

   3. +-commutative 99.1%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)}} \]

   4. log1p-undefine 99.1%

      \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)}} \]

   5. rem-exp-log 99.5%

      \[\leadsto -1 + \color{blue}{\left(1 + \mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} \]

   6. fma-define 99.5%

      \[\leadsto -1 + \left(1 + \color{blue}{\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776} + 0.5\right)}\right) \]

   7. +-commutative 99.5%

      \[\leadsto -1 + \left(1 + \color{blue}{\left(0.5 + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}\right) \]

   8. associate-+r+ 99.5%

      \[\leadsto -1 + \color{blue}{\left(\left(1 + 0.5\right) + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)} \]

   9. metadata-eval 99.5%

      \[\leadsto -1 + \left(\color{blue}{1.5} + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right) \]

10. Simplified 99.7%

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

    1. log1p-expm1-u 99.7%

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log u1 \cdot -0.05555555555555555\right)\right)}}, \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right), 0.5\right) \]

    2. expm1-undefine 99.6%

       \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(\color{blue}{e^{\log u1 \cdot -0.05555555555555555} - 1}\right)}, \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right), 0.5\right) \]

    3. exp-to-pow 99.7%

       \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(\color{blue}{{u1}^{-0.05555555555555555}} - 1\right)}, \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right), 0.5\right) \]

12. Applied egg-rr 99.7%

    \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\mathsf{log1p}\left({u1}^{-0.05555555555555555} - 1\right)}}, \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right), 0.5\right) \]

13. Final simplification 99.7%

    \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{log1p}\left({u1}^{-0.05555555555555555} + -1\right)}, \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right), 0.5\right) \]
14. Add Preprocessing

Alternative 2: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[Log[N[Power[u1, -0.05555555555555555], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(u2 * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. pow1/2 99.4%

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

   2. add-log-exp 99.4%

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

   3. metadata-eval 99.4%

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

   4. *-un-lft-identity 99.4%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   5. metadata-eval 99.4%

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

   6. log-prod 99.4%

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

   7. metadata-eval 99.4%

      \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   8. add-log-exp 99.4%

      \[\leadsto \left(0 + \color{blue}{\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   9. metadata-eval 99.4%

      \[\leadsto \left(0 + \color{blue}{0.16666666666666666} \cdot \sqrt{-2 \cdot \log u1}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\left(0 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

   1. +-lft-identity 99.4%

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

   2. *-commutative 99.4%

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

6. Simplified 99.4%

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

8. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} - 1} \]
9. Step-by-step derivation

   1. sub-neg 99.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} + \left(-1\right)} \]

   2. metadata-eval 99.1%

      \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} + \color{blue}{-1} \]

   3. +-commutative 99.1%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)}} \]

   4. log1p-undefine 99.1%

      \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)}} \]

   5. rem-exp-log 99.5%

      \[\leadsto -1 + \color{blue}{\left(1 + \mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} \]

   6. fma-define 99.5%

      \[\leadsto -1 + \left(1 + \color{blue}{\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776} + 0.5\right)}\right) \]

   7. +-commutative 99.5%

      \[\leadsto -1 + \left(1 + \color{blue}{\left(0.5 + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}\right) \]

   8. associate-+r+ 99.5%

      \[\leadsto -1 + \color{blue}{\left(\left(1 + 0.5\right) + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)} \]

   9. metadata-eval 99.5%

      \[\leadsto -1 + \left(\color{blue}{1.5} + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right) \]

10. Simplified 99.7%

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

    1. add-log-exp 99.6%

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

    2. exp-to-pow 99.7%

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

12. Applied egg-rr 99.7%

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

13. Final simplification 99.7%

    \[\leadsto \mathsf{fma}\left(\sqrt{\log \left({u1}^{-0.05555555555555555}\right)}, \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right), 0.5\right) \]
14. Add Preprocessing

Alternative 3: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(-0.05555555555555555 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(u2 * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. pow1/2 99.4%

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

   2. add-log-exp 99.4%

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

   3. metadata-eval 99.4%

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

   4. *-un-lft-identity 99.4%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   5. metadata-eval 99.4%

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

   6. log-prod 99.4%

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

   7. metadata-eval 99.4%

      \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   8. add-log-exp 99.4%

      \[\leadsto \left(0 + \color{blue}{\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   9. metadata-eval 99.4%

      \[\leadsto \left(0 + \color{blue}{0.16666666666666666} \cdot \sqrt{-2 \cdot \log u1}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\left(0 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

   1. +-lft-identity 99.4%

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

   2. *-commutative 99.4%

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

6. Simplified 99.4%

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

8. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} - 1} \]
9. Step-by-step derivation

   1. sub-neg 99.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} + \left(-1\right)} \]

   2. metadata-eval 99.1%

      \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} + \color{blue}{-1} \]

   3. +-commutative 99.1%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)}} \]

   4. log1p-undefine 99.1%

      \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)}} \]

   5. rem-exp-log 99.5%

      \[\leadsto -1 + \color{blue}{\left(1 + \mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} \]

   6. fma-define 99.5%

      \[\leadsto -1 + \left(1 + \color{blue}{\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776} + 0.5\right)}\right) \]

   7. +-commutative 99.5%

      \[\leadsto -1 + \left(1 + \color{blue}{\left(0.5 + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}\right) \]

   8. associate-+r+ 99.5%

      \[\leadsto -1 + \color{blue}{\left(\left(1 + 0.5\right) + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)} \]

   9. metadata-eval 99.5%

      \[\leadsto -1 + \left(\color{blue}{1.5} + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right) \]

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{-0.05555555555555555 \cdot \log u1} \end{array} \]

FPCore

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

C

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

Java

public static double code(double u1, double u2) {
	return 0.5 + (Math.cos((u2 * (2.0 * Math.PI))) * Math.sqrt((-0.05555555555555555 * Math.log(u1))));
}

Python

def code(u1, u2):
	return 0.5 + (math.cos((u2 * (2.0 * math.pi))) * math.sqrt((-0.05555555555555555 * math.log(u1))))

Julia

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

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + (cos((u2 * (2.0 * pi))) * sqrt((-0.05555555555555555 * log(u1))));
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[Cos[N[(u2 * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-0.05555555555555555 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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. pow1/2 99.4%

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

   2. metadata-eval 99.4%

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

   3. rem-square-sqrt 99.1%

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

   4. sqrt-unprod 99.4%

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

   5. metadata-eval 99.4%

      \[\leadsto \sqrt{\left(\color{blue}{\frac{1}{6}} \cdot \sqrt{-2 \cdot \log u1}\right) \cdot \left(0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   6. metadata-eval 99.4%

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

   7. *-commutative 99.4%

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

   8. *-commutative 99.4%

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

   9. swap-sqr 99.4%

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

   10. add-sqr-sqrt 99.7%

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

   11. metadata-eval 99.7%

       \[\leadsto \sqrt{\left(-2 \cdot \log u1\right) \cdot \left(\color{blue}{0.16666666666666666} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   12. metadata-eval 99.7%

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

   13. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\sqrt{\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. metadata-eval 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

   \[\leadsto 0.5 + \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{-0.05555555555555555 \cdot \log u1} \]
8. Add Preprocessing

Alternative 5: 0.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + \sqrt{\log u1} \cdot \sqrt{-0.05555555555555555} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[Sqrt[N[Log[u1], $MachinePrecision]], $MachinePrecision] * N[Sqrt[-0.05555555555555555], $MachinePrecision]), $MachinePrecision]), $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. pow1/2 99.4%

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

   2. add-log-exp 99.4%

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

   3. metadata-eval 99.4%

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

   4. *-un-lft-identity 99.4%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   5. metadata-eval 99.4%

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

   6. log-prod 99.4%

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

   7. metadata-eval 99.4%

      \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   8. add-log-exp 99.4%

      \[\leadsto \left(0 + \color{blue}{\frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   9. metadata-eval 99.4%

      \[\leadsto \left(0 + \color{blue}{0.16666666666666666} \cdot \sqrt{-2 \cdot \log u1}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\left(0 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

   1. +-lft-identity 99.4%

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

   2. *-commutative 99.4%

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

6. Simplified 99.4%

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

8. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} - 1} \]
9. Step-by-step derivation

   1. sub-neg 99.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} + \left(-1\right)} \]

   2. metadata-eval 99.1%

      \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} + \color{blue}{-1} \]

   3. +-commutative 99.1%

      \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)}} \]

   4. log1p-undefine 99.1%

      \[\leadsto -1 + e^{\color{blue}{\log \left(1 + \mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)}} \]

   5. rem-exp-log 99.5%

      \[\leadsto -1 + \color{blue}{\left(1 + \mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right), \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)\right)} \]

   6. fma-define 99.5%

      \[\leadsto -1 + \left(1 + \color{blue}{\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776} + 0.5\right)}\right) \]

   7. +-commutative 99.5%

      \[\leadsto -1 + \left(1 + \color{blue}{\left(0.5 + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}\right) \]

   8. associate-+r+ 99.5%

      \[\leadsto -1 + \color{blue}{\left(\left(1 + 0.5\right) + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)} \]

   9. metadata-eval 99.5%

      \[\leadsto -1 + \left(\color{blue}{1.5} + \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right) \]

10. Simplified 99.7%

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

11. Taylor expanded in u2 around 0 0.0%

    \[\leadsto \color{blue}{0.5 + \sqrt{\log u1} \cdot \sqrt{-0.05555555555555555}} \]
12. Step-by-step derivation

    1. +-commutative 0.0%

       \[\leadsto \color{blue}{\sqrt{\log u1} \cdot \sqrt{-0.05555555555555555} + 0.5} \]

13. Simplified 0.0%

    \[\leadsto \color{blue}{\sqrt{\log u1} \cdot \sqrt{-0.05555555555555555} + 0.5} \]

14. Final simplification 0.0%

    \[\leadsto 0.5 + \sqrt{\log u1} \cdot \sqrt{-0.05555555555555555} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(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))